# AI tutor | The No.1 Homework Finishing Free App

## 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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.02

"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."

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.05

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.07

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.08

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.09

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.10

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.11

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.12

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.14

'Important Example 118 Region Represented by Logarithmic Inequality'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.15

'Proof of multivariable inequalities related to sequences'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.17

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.18

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.19

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.21

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.22

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.24

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.25

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.26

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.27

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.28

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.29

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.30

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.31

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.32

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.33

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.34

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.36

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.37

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.39

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.40

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.42

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.44

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.45

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.46

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.49

'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?'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.50

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.52

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.53

'Determining coefficients from the solutions of a quadratic inequality'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.55

'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}'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.56

'Text representation of the solution to a quadratic inequality'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.59

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.60

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.61

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.63

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.65

'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.\'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.66

"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."

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.67

'Applications of Systems of Inequalities (Quadratic)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.68

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.71

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.73

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.76

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.77

'Extract information from the solutions of two inequalities.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.79

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.80

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.81

'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. '

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.82

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.83

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.84

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.86

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.87

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.88

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.89

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.90

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.91

'Summary of solution methods for two inequalities'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.96

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.98

'Explain the basic properties of linear inequalities.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.99

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.00

'Practice 3: Application of Simultaneous Second Order Inequalities'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.01

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.02

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.04

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.07

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.08

'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 = □.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.09

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.11

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.13

'Practice 1: Inequality with Absolute Value and Graph'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.15

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.19

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.20

'Summary of methods for solving quadratic inequalities'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.21

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.22

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.25

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.26

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.31

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.32

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.33

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.35

'Prove that the following inequalities hold true.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.36

'Plot the region represented by the following inequality.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.40

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.41

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.42

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.43

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.45

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.46

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.47

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.48

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.49

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.50

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.51

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.52

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.53

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.54

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.55

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.58

'Find the range of values for absolute inequalities'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.59

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.60

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.61

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.62

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.63

'Method for solving inequalities with absolute values'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.64

'[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 \'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.66

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.67

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.68

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.69

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.72

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.73

'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 $x+9 \\leqq 3-5 x \\leqq 2(x-2)$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.74

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.77

'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 \'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.79

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.80

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.81

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.82

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.84

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.85

'Practice stating the negations of the following propositions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.86

'[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 \'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.88

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.93

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.95

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.96

'Practice solving the following quadratic inequalities.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.99

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.00

'Practice solving the following system of linear inequalities.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.01

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.02

'Please explain the basic properties of linear inequalities.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.04

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.05

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.06

'Methods for solving inequalities involving trigonometric ratios'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.08

'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)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.09

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.10

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.13

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.15

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.16

'Solve the inequality containing an absolute value.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.18

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.20

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.25

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.26

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.27

"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)."

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.28

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.30

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.33

'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 $f(a+1)=a^{2}-7 a-9$'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.36

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.37

'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. \\)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.39

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.43

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.44

'Translate the given text into multiple languages.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.47

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.48

'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)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.49

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.50

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.51

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.53

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.54

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.56

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.62

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.66

'Extension of the proof of the Rain 31 inequality'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.68

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.69

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.70

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.71

'Plot the region represented by the following inequalities.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.72

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.73

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.74

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.75

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.76

'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]'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.78

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.79

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.81

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.82

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.83

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.85

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.87

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.88

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.89

'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 $x$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.90

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.91

'Determine whether the following statements are propositions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.93

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.94

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.96

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.97

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.98

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.99

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.00

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.01

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.02

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.04

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.05

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.06

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.07

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.08

'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'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.09

'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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.11

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.12

'(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.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.15

'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 $9-7 x<3-5 x<2(x-2)$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.16

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.17

'State the negation of the following propositions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.19

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) $x^{2}-8x+16 \leqq 0$

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.20

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) $x^{2}+4 x+6 \leqq 0$

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.21

Quadratic Inequality Basics 93 Solving Quadratic Inequalities (1)

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.22

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.23

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.24

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.25

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.

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.26

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.27

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.28

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.29

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.30

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) $x^{2}+3 \geqq 0$

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.31

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.32

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.

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.33

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.34

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) \( 0.2 x-7.1 \leqq-0.5(x+3) \)

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.35

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) $0.1x + 0.06 < 0.02x + 0.1$

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.36

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.37

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.

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.38

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) $|x|>2 \Longrightarrow 3 x+1 \leqq 0$

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.39

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.40

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.

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.41

Quadratic Inequality Basics 94 Solution of Quadratic Inequality (2)

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.42

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.43

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.44

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) $-x^{2}+3 x+2>0$

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.46

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.47

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).

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.48

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.50

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) $-9 x^{2}+12 x-4 \geqq 0$

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.51

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.54

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

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.55

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