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## Numbers and Algebra

### Fundamental Algebra - Solving Inequalities

#### Q.01

'Practice proving that for any positive real numbers $a, b, x, y, z$, the following inequalities hold true and determine the conditions for equality to occur.'

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"Solving the inequality log_{x} y+2・\\frac{1}{log_{x} y}<3. Let's assume log_{x} y=t. Then, we get \\frac{t^{2}-3t+2}{t}<0. Further, \\frac{(t-1)(t-2)}{t}<0, and we solve for t>0 and t<0 cases."

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'Find the range of values for the constant k such that for any real numbers x, y that satisfy the inequalities x^{2}+y^{2}-2x-2y ≤ 0 and x-2y+1 ≤ 0, the inequality y-kx-k-1 ≤ 0 is always satisfied.'

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'Prove the following inequalities:'

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'Find the range of values \u200b\u200bfor the constant a, so that the inequality 3a^2x-x^3≤16 always holds for x≥0.'

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'When x is 29. In other words, we just need to find the conditions that make inequality (2) hold. Therefore, from (2), we have a=c and b=d. Exercise 9|II| => Main book p.52'

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'Solve the following inequalities: (1) 2sinθ - √2 ≥ 0 (2) 2cosθ - 1 < 0 (3) √3tanθ - 1 < 0'

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'Prove the following inequalities. Also, explain when the equality holds.'

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'Solve the inequality $x^{2}+y^{2}<x+y$, plot the region, and make sure it satisfies the other constraint $0 < x + y < 2$.'

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'Find the range of real numbers a such that the inequality 4t^2+at+1-a>0 always holds true when t>0.'

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'Important Example 118 Region Represented by Logarithmic Inequality'

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'Proof of multivariable inequalities related to sequences'

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'Find the region represented by the inequality: (1) Boundary line is a straight line'

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'Find the region represented by the system of inequalities.'

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'When the three inequalities x-y ≥ -2, x-4y ≤ 1, 2x+y ≤ 5 are satisfied simultaneously, find the range of possible values for x+y.'

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'Region represented by the inequality'

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'Prove the following inequalities hold true when a>0, b>0, c>0, d>0. Also, determine the conditions for equality to hold.'

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'Prove the following inequality and determine when the equality holds.'

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'When x>1, find the minimum value of x+1/(x-1).'

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'Inequality proof (3)... using (real numbers)^2 ≥ 0 [Part 2]'

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'Find the region represented by the inequality: (2) The boundary line is a circle'

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'For 0 ≤ x < 2π, find the range of values for x that satisfy the following inequality.'

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'Inequality proof (4)... using the relationship between squares'

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'Prove that the inequality $x^{3}+5>3 x^{2}$ holds.'

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'Find the range of values for x + y when x and y satisfy the three inequalities x - y ≥ -2, x - 4y ≤ 1, 2x + y ≤ 5 simultaneously.'

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'Find the maximum and minimum values of x+y when the inequality 0≤y≤-1/2|x|+3 holds, and the corresponding values of x and y.'

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'Inequality proof (5)...inequality involving absolute value'

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'Prove the following inequalities hold true when a > 0, b > 0. Also, determine the cases where equality holds. (1) a+9/a ≥ 6 (2) 6b/a + 2a/3b ≥ 4'

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'Prove that the inequality a+\\frac{1}{4a} \\geqq 1 holds true when a>0. Also, determine the conditions under which equality holds.'

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'Please solve the following inequality: a+8 > \\frac{3 a}{a+1}'

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'Solve the given system of equations'

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'Solve the following equations and find the range of values for a.'

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'Prove that the inequality e^x > 1 + x holds true, where x is not equal to 0.'

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'Solve the following inequalities: (1) 4 x + 5 > 3 x - 2 (2) 9 - x ≤ 2 x - 3 (3) \\frac{4 - x}{2} > 7 + 2 x'

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'What color is your hat? - The use of reductio ad absurdum -'

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'Quadratic inequalities containing parameters'

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'Conditions for the inequality to hold true at all times'

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'Solution to Quadratic Inequalities (1)'

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'Solve the following inequality: 0.2x - 7.1 > -0.5(x+3)'

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'What is the solution to the inequality |3x-6|<b? If the graph of y=b is above the graph of y=|3x-6|, what range of x values will it cover? Please explain for the case when b>0.'

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'I have already learned that when α<β, the solution for (x−α)(x−β)<0 is that α<x<β. Now we are looking for the reverse scenario.'

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'Solve the following equations and inequalities.'

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'Solve the following inequalities.'

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'What is the solution to the inequality |3x-6|<ax? If the graph of y=ax is above the graph of y=|3x-6|, what range of x values does it correspond to?'

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'Solve the following equations and inequalities: (1) |2x-3|=5 (2) |x-3|>2 (3) 3|1-x|≤2'

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'Solve these two quadratic inequalities.'

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'Determine the values of the constants a and b so that the solution to the quadratic inequality ax^2+9x+2b>0 is 4<x<5.'

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'Determining coefficients from the solutions of a quadratic inequality'

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'Solve the following inequalities. (1) { 4 x + 1 < 3 x - 1 \\ 2 x - 1 ≥ 5 x + 6 } (2) { 2 x + 3 > x + 2 \\ 3 x > 4 x + 2 } (3) 2(x - 3) + 5 < 5 x - 6 ≤ \\frac{3 x + 4}{3}'

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'Text representation of the solution to a quadratic inequality'

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'Solution of second-degree inequalities (2)'

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'Solve the following quadratic inequalities.'

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'Solve the following two inequalities. (2) 6 x^{2}-5 x+1>0'

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'Transformed the inequality (A) in the following (1), (2) into equations (1), (2), (3) sequentially, and derived (3) as the solution to (A). Determine whether the solution (3) is correct or not. If incorrect, state which transformation (A)→(1), (1)→(2), (2)→(3) is incorrect. Here, a is a constant real number. (1) (A): \\sqrt{2} x+3＞2 x+1 、 (1): (\\sqrt{2}-2) x＞-2 、 (2): x＞\\frac{-2}{\\sqrt{2}-2} 、 (3): x＞\\sqrt{2}+2 (2) (A): a^{2}|x|-1＜a^{2}-|x| 、 (1): (a^{2}+1)|x|＜a^{2}+1 、 (2): |x|＜1 、 (3): -1＜x＜1'

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'Solve the inequality: (4) \ -\\frac{2 x+1}{6}<\\frac{x+1}{2} \.'

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'Investigate whether the following two inequalities hold true for all real numbers x:\n(1) 2x^2-3x+1 < 0\n(2) 2x^2-4x+2 ≥ 0\n(3) 2x^2-5x+4 ≤ 0\n(4) 2x^2-6x+4 > 0'

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'Application of two inequalities (solutions:)'

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'Solve the following inequalities:\n(1) \\\left\\{\egin{\overlineray}{l}x^{2}>1+x \\\\ x\\leqq 15-6 x^{2}\\end{\overlineray}\\right.\\n(2) \2\\leqq x^{2}-x\\leqq 4 x-4\\n(3) \\\left\\{\egin{\overlineray}{l}x^{2}+3 x+2\\leqq 0 \\\\ x^{2}-x-2<0\\end{\overlineray}\\right.\'

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"In Hanako's class, she thought about the inequality |3x-6|<ax+b in her math class, where a, b are constants. Let's first find the solution to the inequality (1) when a=2, b=1."

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'Applications of Systems of Inequalities (Quadratic)'

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'Solve the following inequality: (4) - (2x+1)/6 < (x+1)/2 < (1/4)x + 1/3'

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'Solve the equation x^{2}-x-6 \\geqq 0.'

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'Solve the inequality: 5(x-3) < 3(2x-5)'

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'Solve the following equations:\n(1) |3x+8|=5x\n(2) |x+1|+|x-1|=2x+8'

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'Solve the following inequalities.'

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'Solve the inequality (5) |5x-9| ≤ 3 and find the range of x.'

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'Solution: The value of x is [2, 12/5]'

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'Solve the following equations and inequalities.'

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'Examine the relationship between the following 2 numbers and express them as inequalities.'

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'Extract information from the solutions of two inequalities.'

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'Solve the following inequalities.'

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'Examine the truth of the following propositions. Use sets to investigate (2), (3).\n(2) For real numbers x, if |x|>2 then x>2.\n(3) For real numbers x, if |x+2|<1 then |x|<3.'

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'When a = 4, b = 6, find the range of x that satisfies the inequality (1) |3x - 6| < 4x + 6.'

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'Solve the following system of inequalities. (1) \\left\\{ \egin{\overlineray}{l}3x+1>0 \\\\ 3x^{2}+x-10 \\leq 0\\end{\overlineray} \\right. (2) \\left\\{ \egin{\overlineray}{l}x^{2}-4x+1 \\geq 0 \\\\ -x^{2}-x+12>0\\end{\overlineray} \\right. '

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'Solve the inequality \ \\frac{x+1}{2}<\\frac{1}{4} x+\\frac{1}{3} \.'

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'Solve the following inequalities. (3) 3(x+4)/3 - (x-2)/2 > x - 1/6, -2(x-2) < x-5'

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'Solve the following inequality: (3) -x^{2}-x+2 \\geqq 0'

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'Proof of Proposition using Contrapositive'

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'Solve the following inequalities: (1) \\left\\{\egin{\overlineray}{l}2x+3<3x+5\\\\2(x+3)\\le- x+9\\end{\overlineray}\\right. (2) $-4x+1<7-3x<x-1$'

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'Master the properties of inequalities and conquer example 34!'

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"Let's review the basics of simultaneous inequalities and quadratic inequalities!"

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'Chapter 2 Real Numbers, 1st-degree inequalities: 61 inequalities'

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"Let's review the basics of quadratic inequalities!"

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'Summary of solution methods for two inequalities'

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'Explain the properties of a linear inequality.'

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'Solve the following 2 quadratic inequalities:'

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"Let's review the basics of solving systems of inequalities and quadratic inequalities!"

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'Consider the inequality |2x-6|<x'

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'Explain the basic properties of linear inequalities.'

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'Master the properties of inequalities and conquer Example 34!'

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'Practice 3: Application of Simultaneous Second Order Inequalities'

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"Discriminant, let's review the basics of quadratic inequalities!"

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'Solve the following quadratic inequalities. (1) x^2 + 2x + 1 > 0 (2) x^2 + 4x + 4 ≥ 0 (3) 1/4 x^2 - x + 1 < 0 (4) -9x^2 + 12x - 4 ≥ 0'

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'(2) $\\left|x^{2}-6 x-7\\right| \\geqq 2 x+2$'

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'Find the range of values for the constant m such that the given two inequalities hold true.'

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'Summary of solving two inequalities'

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'Solve the inequality 3|x+1|≥x+5.'

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'Find the range of values for the constant a so that the inequality always holds within a certain range'

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'The solution to inequality (1) is y = |2x-6| - x...the range of values of x for which y < 0 on the graph of (2). (Use the graph of (2) to find the solution to (1).\n(i) Solving 2x-6 ≥ 0 gives x ≥ □, solving 2x-6 < 0 gives x < □, therefore when x ≥ □, (2) is y = □.\nWhen x < □, (2) is y = □.'

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'Let TR(x) be a real number. Using sets, determine the truth value of the following propositions.'

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'Solve the following quadratic inequalities.'

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'Solve the system of inequalities { |x+1|<\\frac{3}{2}, x^{2}-2 x-3>0 }.'

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'Solve the following inequalities:'

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'Practice 1: Inequality with Absolute Value and Graph'

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'Solve the following quadratic inequalities.'

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'Determine the values of constants a and b such that:\n(1) The solution to the quadratic inequality x^2 + ax + b < 0 is -\x0crac{1}{2} < x < 3.\n(2) The solution to the quadratic inequality ax^2 + x + b ≤ 0 is x ≤ -1, 2 ≤ x.'

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'Solve the inequality 3|x+1| ≥ x+5.'

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'Solve the inequality |2x|+|x-5|≥8.'

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'Solve the following inequalities: (1) \\left\\{\egin{\overlineray}{l}4 x-1<3 x+5 \\\\ 5-3 x<1-x\\end{\overlineray}\\right.'

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'Summary of methods for solving quadratic inequalities'

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'Find the solution to the system of inequalities \\(\\left\\{\egin{array}{l}x>3a+1\\\\2x-1>6(x-2)\\end{array}\\right.\\) and determine the range of constant \a\ that satisfies the following conditions.\\n1. No solution exists.\\n2. The solution contains 2.\\n3. Only 3 integers are included in the solution.'

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'Solve the inequality p x ≥ 2 x-3 for a constant p.'

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'Solve the following inequalities.'

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'Solve the following quadratic inequalities.'

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'Find the range of values for the constant $a$ that satisfies the following conditions for the solution of the system of inequalities \\left\\{\egin{\overlineray}{l}3 x-7 \\leqq 5 x-3 \\\\ 2 x-6<3 a-x\\end{\overlineray}\\right.: (1) The system has a solution. (2) The solution contains exactly 3 integers.'

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'Find all the values of the integer x that satisfy the system of inequalities {2(x+1) ≥ 5x-2, -5x < -3x+4}.'

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'Solve the following inequalities.'

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'Solve the two inequalities using a graph.'

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'(1) y ≥ 0\n(2) y ≤ 0\n(3) 0 ≤ y ≤ 6\n(4) -1 ≤ y < 1'

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'Prove that 3x+1>x+3 when x>1. (2) Prove the inequality a^2-2ab+3b^2 ≥ 0 and determine the conditions for equality.'

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'Solve the inequality \\\log _{2} x-6 \\log _{x} 2 \\geqq 1 \.'

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'(1) Prove that if x+y>0 and x-y>0, then 2x+y>0.'

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'Prove that the following inequalities hold true.'

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'Plot the region represented by the following inequality.'

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'Prove the following inequalities:'

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'Solve the inequality $2 \\log _{3} x-4 \\log _{x} 27 \\leqq 5$.'

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'What inequality represents the shaded area on the right of region A? Assume it does not include the boundary lines.'

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'Solution to exponential inequalities: Solve the following exponential inequality.'

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'Solution of trigonometric equations and inequalities (substitution of angles)'

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'Solve the inequality 2log₃x - 4logₓ27 ≤ 5.'

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'Solve the following inequalities: (1) $\\left(\\frac{1}{3}\\right)^{x}<9$ (2) $9^{x+1}-28\\cdot 3^{x}+3<0$ (3) $4^{x}-2^{x+2}-32>0$'

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'Prove the following inequalities and determine when the equality holds.'

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'Solution of logarithmic inequalities (2): Solve the following logarithmic inequality.'

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'Find the maximum and minimum values of x-4 y when the inequalities x >= 0, y >= 0, x-2 y+8 >= 0, 3 x+y-18 <= 0 are satisfied.'

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'Prove the inequality $\\frac{3}{10}<\\log_{10} 2<\\frac{4}{13}$ without using $\\log_{10} 2=0.3010 \\cdots \\cdots$. Since $2^{10}=1024,2^{13}=8192$, we have $10^{3}<2^{10}, 2^{13}<10^{4 }$. Taking the common logarithm of both sides of these inequalities.'

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'Solve the inequality: \9^x < 27^{5-x} < 81^{2x+1}\'

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'Find the minimum value of a such that the inequality a sin ^{2} x+6 sin x+1 always holds true.'

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'When PR (x, y) satisfies the 4 inequalities x≥0, y≥0, x-2y+8≥0, 3x+y-18≤0, find the maximum and minimum values of x-4y taken.'

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'Dealt with matters that are not covered in the STEP UP textbook, especially those that require special attention.'

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'Prove that the following inequalities hold when a ≥ 0 and b ≥ 0 in PR. Also, determine the conditions for which equality holds.'

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'Solution to logarithmic inequalities (1): Solve the following logarithmic inequality.'

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'Solve the following equations and inequalities.'

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'Find the range of values for absolute inequalities'

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'Solve the following equations:\n(1) $|x-2|=3x$\n(2) $|x-1|+|x-2|=x$'

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'Solve the inequality a(x+1) > x + a^{2}, where a is a constant.'

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'Find the range of constants m for which the inequality x^2 - 2mx + m + 6 > 0 holds for all values of x in the range 0 ≤ x ≤ 8.'

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'Find the range of values for the constant a such that the inequality a(x^2+x-1) < x^2+x holds for all real numbers x.'

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'Method for solving inequalities with absolute values'

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'[1] When \ x \\leqq-1,7 \\leqq x \, the inequality is\n\\nx^{2}-6 x-7 \\geqq 2 x+2\n\\nTherefore, \ \\quad x^{2}-8 x-9 \\geqq 0 \, hence \\( (x+1)(x-9) \\geqq 0 \\)\nTherefore, \ \\quad x \\leqq-1,9 \\leqq x \'

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'Solve the following inequalities.'

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'For a>0 and D>0, if the two distinct real solutions of the quadratic equation ax^2 + bx + c = 0 are α and β (α<β), then the solutions are x < α or β < x for a x^2 + b x + c > 0. The solutions are α < x < β for a x^2 + b x + c < 0. The solutions are x ≤ α or β ≤ x for a x^2 + b x + c ≥ 0. The solutions are α ≤ x ≤ β for a x^2 + b x + c ≤ 0.'

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'Solve the two inequalities $a(x-3a)(x-a^2)<0$. where $a$ is a non-zero constant.'

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'Solving two inequalities (2) when α<β, if there is an equal sign, it will be included in the solution'

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'Find all the values of the natural number x that satisfy 5x - 7 < 2x + 5.'

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'Find the range of the inequality x^2-4x+1=t.'

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'Properties of linear inequalities: If a < b, then a + c < b + c, a - c < b - c. If a < b and c > 0, then ac < bc, a/c < b/c. If a < b and c < 0, then ac > bc, a/c > b/c. If a < b and b < c, then a < c.'

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'Solve the following systems of inequalities: (1) \\left\\{\egin{\overlineray}{l}2(1-x)>-6-x \\\\ 2 x-3>-9\\end{\overlineray}\\right. (2) \\left\\{\egin{\overlineray}{l}3(x-4) \\leqq x-3 \\\\ 6 x-2(x+1)<10\\end{\overlineray}\\right. (3) Solve the inequality $x+9 \\leqq 3-5 x \\leqq 2(x-2)$.'

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'In equations or inequalities involving absolute values, remove the absolute value and solve the equation or inequality by considering the expression inside the absolute value as a breakpoint.'

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'Solve the following linear inequalities.'

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'Solve the following equations and inequalities:'

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'Practice solving the following equations and inequalities.\n(1) \ |x+5|=3 \\n(2) \ |1-3 x|=5 \\n(3) \ |x+2|<5 \\n(4) \ |2 x-1| \\geqq 3 \'

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'Find the range of values \u200b\u200bfor the constant a where there exist exactly 3 integers x that simultaneously satisfy the inequalities x²-(a+1)x+a≤0 and 3x²+2x-1≥0.'

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'Find the range of constants a for which the inequality ax^2 + y^2 + az^2 - xy - yz - zx ≥ 0 holds for any real numbers x, y, z. Given inequality rearranges in terms of y as y^2 - (z + x)y + a(z^2 + x^2) - zx ≥ 0.'

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'Practice stating the negation of the following propositions and determine the truth value of the original propositions and their negations.'

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'Linear inequality with absolute value (using graphical method)'

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'Practice solving the following inequalities.'

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'Solve the inequality \ \\left|x^{2}-2 x-3\\right| \\geqq 3-x \.'

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'Practice stating the negations of the following propositions.'

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'[1] When \ x \\leqq-1, \\frac{5}{2} \\leqq x \, the inequality is \ 2 x^{2}-3 x-5<x+1 \. Simplifying we get \ \\quad x^{2}-2 x-3<0 \, so \\( \\quad(x+1)(x-3)<0 \\). Therefore, \ \\quad-1<x<3 \. The common range with \ x \\leqq-1, \\quad \\frac{5}{2} \\leqq x \ is \ \\quad \\frac{5}{2} \\leqq x<3 \'

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'Solve the inequality ax > 3x - b.'

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'When a=-1, y is a constant in the interval -1 ≤ x ≤ 2.'

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'Solve the following inequalities.'

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'Solve the following inequalities.'

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'Solve the inequality |2x-3| ≤ |3x+2|.'

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'Let a, b, c, and p be real numbers. Assume that the set of real numbers x satisfying the inequalities ax^2+bx+c>0, bx^2+cx+a>0, and cx^2+ax+b>0 is equal to the set of real numbers x satisfying x>p.'

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'For all real numbers x, y, whether 9x²-12xy+4y²>0'

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'Practice solving the following quadratic inequalities.'

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'Solve the following inequalities:'

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'Determine the range of values for the constant a such that there exists exactly one integer that simultaneously satisfies the two quadratic inequalities x^2 - 2x - 8 < 0 and x^2 + (a - 3)x - 3a ≥ 0.'

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'Practice solving the following system of linear inequalities.'

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'Solve the following equations and inequalities:\n(1) |x-3| + |2x-3| = 9\n(2) ||x-2|-4| = 3x\n(3) |2x-3| ≤ |3x+2|\n(4) 2|x+2| + |x-4| < 15'

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'Please explain the basic properties of linear inequalities.'

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'Solve the inequality 2x^4-11x^2+5>0.'

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'Solve the following linear inequalities:\n(1) 4x - 5 > 3x + 2\n(2) -2x + 7 ≤ 5'

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'For all real numbers x, determine the range of values for the constant a such that the inequality (a-1) x^2 - 2(a-1) x + 3 ≥ 0 holds.'

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'Methods for solving inequalities involving trigonometric ratios'

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'Solve the following quadratic inequalities.'

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'Assume inequality A < B and prove the following inequalities:\n1. A + C < B + C\n2. A - C < B - C\n3. A * C < B * C (where C > 0)\n4. A / C < B / C (where C > 0)'

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'Solutions:\n(1) x > 2\n(2) x > -23\n(3) 11/5 < x < 3\n(4) -2 < x < 0\n(5) -6/5 <= x <= 3\n(6) x < -2 or x > 6'

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'Solve the following inequalities:\n(1) x+3<2\n(2) 2x ≥ 5\n(3) -3x ≤ 4'

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'Solve the inequality \|x^{2}-4| \\leqq x+2\.'

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'Solve the following quadratic inequality. (6) 2 x-3>-x^{2}'

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'Is the solution to inequality (5) x > 0?'

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'Solve the following quadratic inequality: (5) 4 x^{2}-5 x-3<0'

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'Solve the inequality containing an absolute value.'

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'Solve the following equations or inequalities.'

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'When the older brother gives 3 pencils to the younger brother, the younger brother ends up with more'

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'Solve the following inequalities.'

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'Find the conditions for real numbers a and b such that ax^2 + 2bx + 1 > 0 holds for all real numbers x.'

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'Solve the following system of inequalities.'

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'Solve the inequality 2x^{2} - 3x - 5 > 0.'

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'(4) Solve the inequality x^{2}-2 x-2 \\leqq 0'

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'Let a and b be constants, and let x^{2}-5x+6 ≤ 0 (1), x^{2}+ax+b < 0 (2). Given that there are no values of x that satisfy both (1) and (2) simultaneously, and the range of x values that satisfy (1) or (2) is 2 ≤ x < 5. Find the values of a and b.'

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'Solve the inequality | x ^ {2} -2x | > 2-x with absolute value.'

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"Explain the difference between finding the 'common range' and the 'combined range' when solving equations or inequalities that include absolute value symbols. Explain based on example 36 (1)."

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'Solve the following two inequalities:\n(2) 6 x^{2}-5 x+1>0'

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'Solve the following two quadratic inequalities.'

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'Solve the following two quadratic inequalities:\n(3) -x^{2}-x+2 \\geqq 0'

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'Solve the following equations or inequalities.'

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'Solve the following system of inequalities.'

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'When $a+1<5$, i.e., $a<4$, from Figure [4], it can be seen that $x=a+1$ is minimized. The minimum value is $f(a+1)=a^{2}-7 a-9$'

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'Solve the inequality: x(x-1) < 0'

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'Solve the following inequalities.'

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'Find the conditions under which an inequality always holds in a certain range of variables.'

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'Solve the following inequalities. 1. \ \\ left \\ {\\ begin \\ {\overlineray} {l} x ^ {2}> 1 + x \\\\ x \\ leqq 15-6 x ^ {2} \\ end \\ right. \ 2. \\ (2 \\ leqq x ^ {2} -x \\ leqq 4 x-4 \\) 3. \\ (\\ left \\ {\\ begin \\ {array} {l} x ^ {2} + 3 x + 2 \\ leqq 0 \\\\ x ^ {2} -x-2 <0 \\ end \\ right. \\)'

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'Solve the following inequalities.'

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'Determine the condition on the constant a such that there is exactly one integer value of x that satisfies the inequality 2x^{2} - 3x - 5 > 0 and x^{2}+(a-3)x-2a+2<0 simultaneously.'

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'State the negation of the following conditions.'

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'Solve the following inequalities.'

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'Solve the given inequality $2 x^{4}-11 x^{2}+5>0$.'

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'Translate the given text into multiple languages.'

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'Solve the following inequalities.'

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'Solve the following inequalities.'

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'Prove that for all positive numbers x and y, the inequality x(logx - logy) ≥ x - y holds true. Also prove that the equality only holds when x = y.'

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'Find the range of values \u200b\u200bof a for which the following inequality holds for any real number x: 88 (1) a x ≥ log x (x > 0) (2) a^x ≥ x^a (x ≥ a > 0)'

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'Complex numbers and inequalities Let \ \\alpha \ be a complex number with \ |\\alpha|<1 \. When a complex number \ z \ satisfies the inequality \ \\left|\\frac{\\alpha+z}{1+\\overline{\\alpha} z}\\right|<1 \, prove that \ |z|<1 \ holds true.'

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'How to solve the inequality \ 2 x^{2}+x-6 \\geq 0 \?'

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'Prove the following inequalities using the inequality $|a+b| \\leqq |a|+|b|$:'

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'Solve the following inequalities.'

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'Explain the meaning of the following expression: \ p < q \\Leftrightarrow a^p > a^q \, where \ 0 < a < 1 \.'

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'Solve the following inequalities for 0 ≤ θ < 2π.'

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'Solve the following inequality. Where a is a positive constant. \\[x^{3}-(a+1) x^{2}+(a-2) x+2 a \\leqq 0\\]'

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'Proof of 27 inequalities [using A-B>0 etc.]'

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'Prove and find when the equality holds'

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'Graphing logarithmic inequalities and regions'

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'Solution to exponential inequalities'

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'Solve the following inequalities.'

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'Solve the inequality \ \\log _{4} x^{2}-\\log _{x} 64 \\leqq 1 \.'

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'Handling inequalities involving absolute values'

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'Practice solving the following inequalities.'

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'Extension of the proof of the Rain 31 inequality'

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'Region represented by an inequality'

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'Express the sizes of the following sets of numbers using inequality symbols.'

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'Solve the inequality $\\log_{4}x^{2}-\\log_{x}64 \\leqq 1$.'

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'Find the area of the region represented by the system of inequalities y≥x², y≤10-3x, y≤x+6.'

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'Plot the region represented by the following inequalities.'

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'Explain about the region represented by a system of inequalities. In particular, describe cases where the system of inequalities includes the intersection of lines or multiple regions.'

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'Prove that the inequality holds. When does the equality hold?'

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'Find the region represented by the following inequalities: when x<0, 2x ≤ y ≤ 1; when 0 ≤ x < 1/2, 2x ≤ y ≤ x²+1; when 1/2 ≤ x ≤ 1, 1 ≤ y ≤ x²+1; when x > 1, 1 ≤ y ≤ 2x.'

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'Exercise 6\nFor a positive constant a, find the range of values of a such that the inequality a^x >= x holds for all positive real numbers x.'

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'Let K be the region defined by the inequality -sin x ≤ y ≤ cos 2x, 0 ≤ x ≤ π/2. (1) Find the area of K. (2) Find the volume of the solid obtained by rotating K around the x-axis. [Kobe University]'

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'Proof of the inequality f(x)>g(x)'

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'Find the range of values of the positive real number x that satisfy the inequality $x^{2}>2^{x}$.'

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'When two real numbers x, y satisfy the two inequalities y≤x+1 and x²+4y²≤4, find the maximum and minimum values of y-2x.'

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'Prove the inequality |S_{n}-\\int_{0}^{1} \\frac{1}{1+x} d x| \\leqq \\frac{1}{n+1}.'

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'When real numbers x, y satisfy the inequalities y≤2x+1 and x²+2y²≤22, find the maximum and minimum values of x+y.'

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'Prove the inequality \ \\frac{1}{n}+\\log n \\leqq \\sum_{k=1}^{n} \\frac{1}{k} \\leqq 1+\\log n \.'

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'Solve the following inequalities.'

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'Find the range of values of a that make the inequality a^x >= x hold for all positive real numbers x.'

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'Solve the inequality $\\sqrt{4-x}>x-2$.'

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'Prove the inequality $\\left|\\sqrt{x^{2}+(y+1)^{2}}-\\sqrt{y^{2}+(x+1)^{2}}\\right| \\leqq \\sqrt{2}|x-y|$.'

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'Example 130: Maximum and Minimum in a Domain (2)\nLet A be the region represented by the system of inequalities x-2y+3≥0, 2x-y≤0, x+y≥0. Find the maximum and minimum values of y^2-4x as the point (x, y) moves within region A.'

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'Find the range of values for the constant $a$ when the inequalities $x^{2}-10 x-24>0,(x+1)(x-a^{2}+a)<0$ hold simultaneously and there is no existing $x$.'

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'Investigate the truth value of the following propositions and their negations.\n(1) For all real numbers x, 2 x^{2}+1>0\n(2) For any real numbers x, y, x^{2}-4 x y+4 y^{2}>0\n(3) There exists a natural number x such that x^{2}-3 x-10=0'

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'Determine whether the following statements are propositions.'

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'Solve the following equations and inequalities.'

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'Find the range of the inequality 3x - 5x < 4 + 8.'

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'Multiply both sides of the inequality by -1 and solve 2x^2 + 3x + 7 ≤ 0. Let D be the discriminant of the quadratic equation 2x^2 + 3x + 7 = 0, then D = 3^2 - 4 * 2 * 7 = -47 < 0. Therefore, there are no solutions to the given inequality. As an alternative solution, complete the square and verify the lack of solutions.'

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'Solve 3(x-1) \\geqq x+3 and find the range of the solution.'

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'If the number of items to buy is 10 or less, then A store is cheaper than B store, so let the required quantity be x, then x > 10.'

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'Practice\n86\n(1) From \ 2 \\sin \\theta>\\sqrt{2} \, we get \ \\quad \\sin \\theta>\\frac{1}{\\sqrt{2}} \, which leads to \ \\sin \\theta=\\frac{1}{\\sqrt{2}} \ resulting in \ \\quad \\theta=45^{\\circ}, 135^{\\circ} \\n\nThe solution to the inequality lies within the range of values of \ \\theta \ where a point on the circumference of a semicircle has a y-coordinate greater than \ \\frac{1}{\\sqrt{2}} \, hence \ 45^{\\circ}<\\theta<135^{\\circ} \\n\n(2) From \ 2 \\cos \\theta<\\sqrt{3} \, we get\n\n\ \\cos \\theta<-\\frac{\\sqrt{3}}{2} \\n\nSolving \ \\cos \\theta=-\\frac{\\sqrt{3}}{2} \, we find \ \\quad \\theta=150^{\\circ} \\nThe solution to the inequality lies within the range of angles where a point on the circumference of a semicircle has an x-coordinate smaller than \ -\\frac{\\sqrt{3}}{2} \\n1. Pythagorean theorem.\nConsidering an alternative solution where a perpendicular DE is dropped from point D to side \ \\mathrm{AB} \, E becomes the midpoint of side \ \\mathrm{AB} \\n\n\ \egin{aligned}\n\\mathrm{AE} & =\\frac{\\sqrt{4+2 \\sqrt{2}}}{2} \\\\\n\\mathrm{DE} & =\\sqrt{\\mathrm{AD}^{2}-\\mathrm{AE}^{2}} \\\\\n& =\\frac{\\sqrt{4-2 \\sqrt{2}}}{2}\n\\end{aligned} \\n\nTherefore,\n\ \\sin 22.5^{\\circ}=\\frac{\\mathrm{DE}}{\\mathrm{AD}} \,\n\ \\cos 22.5^{\\circ}=\\frac{\\mathrm{AE}}{\\mathrm{AD}} \\n can also be determined.\n4. Practice\nFigure\nShape\nCalculate\nYour Range\nWhen dealing with inequalities involving trigonometric ratios, first set the inequality to = and solve the trigonometric equation. \ \\leqq 180^{\\circ} \ so the angle range should be within \ 150^{\\circ}<\\theta \\leqq 180^{\\circ} \ without exceeding 180 degrees.'

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'Solve the following equations and inequalities. (1) |x-4|=3x (2) 2|x-1|=x+2 (3) 2|x|+|2x+3|=7 (4) |3x-4|<2x (5) 3|x-1|>=x+3 (6) 3|x-2|-2|x|<=3'

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'Find the range of real numbers x that satisfy the following quadratic inequality: a x^2 - 3x + b > 0'

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'Let a be a constant. Find the range of values of a for which there exist integers that simultaneously satisfy the following two inequalities, and these integers are limited to natural numbers.'

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'Let a be a constant. Solve the following inequalities for x.'

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'Prove the inequality |x+2|-|x-1| <= 3.'

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'Practice 69 Booklet p.146 (1) Inequality 2(x-1/2)(x-(a+1))<0 [1] 1/2<a+1 i.e. a>-1/2, when this condition is met, the solution to the inequality is 1/2<x<a+1. The condition for there to be only one integer x satisfying this inequality is 1<a+1 ≤ 2, hence 0<a≤ 1 [2] 1/2=a+1 i.e. a=-1/2, in this case the inequality becomes 2(x-1/2)^{2}<0, and no integer x satisfies this condition. [3] a+1<1/2 i.e. a<-1/2, in this case the solution to the inequality is a+1<x<1/2. The condition for there to be only one integer x satisfying this inequality is -1 ≤ a+1 <0, hence -2≤ a<-1. Therefore, the range of the required values for a is -2 ≤ a<-1, 0<a ≤ 1'

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'Let a, b, c, and p be real numbers. It is given that the set of all real numbers x satisfying the inequalities ax^2+bx+c>0, bx^2+cx+a>0, and 7cx^2+ax+b>0 is equal to the set of real numbers x satisfying x>p.'

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'-3(x-2) + 2x \\leqq 3 and 3(x-2) - 2x \\leqq 3 are solved, and the solution range is requested.'

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'When x = 1, the inequality becomes 0 > 0, which is not a solution.'

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'Find the common range of the following inequalities.\n(A): 6x+5≥2x-3 and x+13 > 7x-5\n(B): 2x+8 > x+7 and 3x-3 > 4x-1'

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'Determine the values of constants a and b so that the following conditions are met:\n(1) The solution to the quadratic inequality ax^2 + 8x + b < 0 is x<-1, 5<x.\n(2) The solution to the quadratic inequality 2x^2 - ax + 3 < 0 is b<x<3.'

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'Solve the following equations and inequalities.'

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'Answer the following questions regarding the two inequalities ||x-9|-1|≤2 (1), |x-4|≤k (2), where k is a positive constant.'

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'(1) Since x^2 >= 0, we have 2x^2 + 1 > 0. Given △ A ∩ B ⊂ A and 2 ∈ A, we consider the case where 7 ∈ A. With 2 ∈ B and 7 ∉ B. (2) To show A = B, we need to demonstrate A ⊂ B and B ⊂ A.'

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'[3] -a/2 > 1 that is, when a < -2'

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'Solve the following inequalities.'

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'Side 17 | Method for solving simultaneous inequalities\n(1) Solve the following simultaneous inequalities.\n(a) \\left\\{\egin{\overlineray}{l}6 x+5 \\geqq 2 x-3 \\\\ x+13>7 x-5\\end{\overlineray}\\right. \n(b) \\left\\{\egin{\overlineray}{l}2 x+8>x+7 \\\\ 3 x-3>4 x-1\\end{\overlineray}\\right. \n(2) Solve the inequality $9-7 x<3-5 x<2(x-2)$.'

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'Solve -2 x < 3 x - 4 < 2 x, which is equivalent to |3 x-4| < 2 x, to find the solution.'

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'State the negation of the following propositions.'

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'Solve the following inequality: |x-1|+2|x| <= 3'

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Solve the following quadratic inequalities.
(1) $x^{2}-8x+16>0$
(2) $x^{2}-8x+16<0$
(3) $x^{2}-8x+16 \geqq 0$
(4) $x^{2}-8x+16 \leqq 0$

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Solve the following quadratic inequalities.
(1) $x^{2}+4 x+6>0$
(2) $x^{2}+4 x+6<0$
(3) $x^{2}+4 x+6 \geqq 0$
(4) $x^{2}+4 x+6 \leqq 0$

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Quadratic Inequality Basics 93 Solving Quadratic Inequalities (1)

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Solve the following quadratic inequalities.
(1) $2 x^{2}-5 x+2<0$
(2) $-4 x^{2}+4 x+1 \leqq 0$

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Basic 32 Using the Properties of Inequalities (1)
Standard 33 Using the Properties of Inequalities (2)
Basic 34 Solving Linear Inequalities
Basic 35 Solving Systems of Inequalities
Standard 36 Integer Solutions of Inequalities
Standard 37 Inequality Word Problems
Basic 38 Equations and Inequalities Involving Absolute Values…Basic
Standard 39 Solving Equations Involving Absolute Values by Case Analysis

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Solve the following equations and inequalities.
(1) \( |(\sqrt{14}-2) x+2|=4 \)
(2) $3|x-1| \leqq 4$
(3) $x+|3 x-2|=3$

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Find the range of values for the constant $a$ that satisfy the following conditions for the solution of the system of inequalities \( \left\{egin{array}{l}x>3a+1 \ 2x-1>6(x-2)\end{array}
ight. \):
(1) No solutions exist.
(2) The solution includes 2.
(3) The solution contains exactly 3 integers.

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Exercise problem 93 - Solve the following quadratic inequality. (6) $x^{2}+x \leqq 6$

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Practice Problem 93 - Solve the following quadratic inequality. (4) $x^{2}+6x+8 \geqq 0$

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Determine the truth value of the propositions. Let $x$ be a real number and $n$ be an integer. Use sets to investigate the truth of the following propositions.
(1) $x<-3 \Longrightarrow 2 x+4 \leqq 0$
(2) $n$ is a positive divisor of 18 $\Longrightarrow n$ is a positive divisor of 24

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Solve the following quadratic inequality. (3) $x^{2}-3 x-10 \leqq 0$

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Solve the following quadratic inequalities.
(1) $x^{2}-4 x+5<0$
(2) $2 x^{2}-8 x+13>0$
(3) $3 x^{2}-6 x+6 \leqq 0$
(4) $x^{2}+3 \geqq 0$

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Find all integer values of $x$ that satisfy the system of inequalities \( \left\\{egin{array}{l}2(x+1) \\geqq 5x-2 \\ -5x<-3x+4\end{array}\right\\} \).

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Find the range of values for the constant $a$ that satisfy the following conditions for the solution of the system of inequalities:
(1) There is a solution.
(2) The solution contains exactly 3 integers.
\left\{egin{\overlineray}{l}3 x-7 \leqq 5 x-3 \\ 2 x-6 < 3 a-x\end{\overlineray}\right.

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Practice Problem 93 - Solve the following quadratic inequality. (3) $x^{2}-2x < 0$

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Solve the following inequalities.
(1) $8 x+13>5 x-8$
(2) \( 3(x-3) \geqq 5(x+1) \)
(3) rac{4-x}{2}<7+2 x
(4) \( rac{1}{2}(1-3 x) \geqq rac{2}{3}(x+7)-5 \)
(5) \( 0.2 x-7.1 \leqq-0.5(x+3) \)

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Solve the following inequalities.
(1) $4x + 5 > 2x - 3$
(2) \( 3(x - 2) \geq 2(2x + 1) \)
(3) rac{1}{2}x > rac{4}{5}x - 3
(4) $0.1x + 0.06 < 0.02x + 0.1$

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Given \mathbf{4 3}^{\Perp}-rac{5}{2} \leqq x \leqq 2 , find the maximum value of the function \( f(x)=(1-x)|x+2| \).

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For the given system of inequalities \left\{egin{\overlineray}{l}x<6 \\ 2 x+3 \geqq x+a\end{\overlineray}\right. , find the range of values for the constant $a$ that satisfy the following conditions:
(1) The system has a solution.
(2) The solution includes exactly 2 integers.

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Let $x$ be a real number. Use sets to determine the truth value of the following propositions.
(1) $-1<x<1 \Longrightarrow 2 x-2<0$
(2) $|x|>2 \Longrightarrow 3 x+1 \leqq 0$

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Practice problem 93 - Solve the following quadratic inequality. (5) $x^{2} > 9$

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Let's organize the method of solving quadratic inequalities in the form of a flowchart. In basic examples 93 to 96, we solved various types of quadratic inequalities. Now we'll organize them.
Consider $ax^{2} + bx + c$. First, check if it can be factored. If it can, it will take the form of \( a(x-lpha)(x-eta) \) or \( a(x-lpha)^{2} \).
Next, draw the graph $y = ax^{2} + bx + c$, assuming $a > 0$.
The summary is as follows:
egin{\overlineray}{c}
Type of Inequality & Range of Solutions \\hline \( a x^{2}+b x+c>0 & x<lpha, eta<x \\hline $a x^{2}+b x+c \geqq 0$ & x \leqq lpha, \quad eta \leqq x \\hline $a x^{2}+b x+c<0$ & lpha<x<eta \\hline $a x^{2}+b x+c \leqq 0$ & lpha \leqq x \leqq eta \\hline\end{array}\)
It is important to understand the reason for the solutions rather than simply memorizing this table.

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Quadratic Inequality Basics 94 Solution of Quadratic Inequality (2)

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Exercise 93 - Solve the following quadratic inequality. (2) \( (2x+1)(3x-5) > 0 \)

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Solve the inequality 2|x+4|<x+10 by considering different cases.

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Solve the following quadratic inequalities.
(1) $3 x^{2}+10 x-8>0$
(2) $6 x^{2}+x-12 \leqq 0$
(3) $5 x^{2}+6 x-1 \geqq 0$
(4) \( 2(x+2)(x-2) \leqq(x+1)^{2} \)
(5) $-x^{2}+3 x+2>0$

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Solve the inequality $x^2-4x+3<0$.

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Exercise 93 - Solve the following quadratic inequality. (1) \( (x+2)(x+3) < 0 \)

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When all terms of the inequality are moved to the left side and simplified, such as $a x^{2}+b x+c>0$ and $a x^{2}+b x+c \leqq 0$ (where $a, b, c$ are constants and a
eq 0 ）, the left side becomes a quadratic expression in \( x , which is called a quadratic inequality in $x$. Then, the value of $x$ that satisfies the quadratic inequality is called the solution of the inequality, and finding all solutions is called solving the quadratic inequality. In this section, let’s learn to solve quadratic inequalities using the graphs of quadratic functions. We will actively make use of the relationship between the graph of a quadratic function and the $x$ axis that we learned in the previous section.
■ Solutions of Quadratic Inequalities and Graphs of Quadratic Functions
Note: In the following discussions, it is not inconvenient to proceed by assuming the sign of the coefficient $a$ of $x^{2}$ to be positive. This is because when the sign of $a$ is negative, we can simply multiply both sides by -1 to make the coefficient of $x^{2}$ positive and then solve. To find the solutions of the quadratic inequality \( a x^{2}+b x+c>0 (a>0) \), let $y = a x^{2}+b x+c$, then finding the range of $x$ for which $y>0$ means finding the range of $x$ where the graph of $y=a x^{2}+b x+c$ is above the $x$ axis.
As a concrete example, let’s explain the quadratic inequality $x^{2}-4 x+3>0$(1).

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Solve the following quadratic inequality. (2) \( (x+1)(x-2) < 0 \)

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Solve the inequality 3|x+1| $\geqq x+5$.

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Solve the following quadratic inequalities.
(1) $x^{2}+2 x+1>0$
(2) $x^{2}+4 x+4 \geqq 0$
(3) rac{1}{4} x^{2}-x+1<0
(4) $-9 x^{2}+12 x-4 \geqq 0$

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Express the statement 'The number obtained by subtracting three times a number x from 10 is greater than -5' as an inequality.

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Solve the inequality $|2 x|+|x-5| \geqq 8$.

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Solve the inequality $x^2 - 4x + 3 > 0$.

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Solve the following inequalities.
(1) \left\{egin{\overlineray}{l}4 x-1<3 x+5 \ 5-3 x<1-x\end{\overlineray}
ight.
(2) \left\{egin{\overlineray}{l}3 x-5<1 \ rac{3 x}{2}-rac{x-4}{3} \leqq rac{1}{6}\end{\overlineray}
ight.
(3) rac{2 x+5}{4}<x+2 \leqq 17-2 x

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Solve the following equations and inequalities.
(1) $|2 x-1|=7$
(2) $|2 x+5| \leqq 2$
(3) $|3-x|>4$