Master Linear Inequalities: A Comprehensive Guide for High School Sophomores

Question 1

Solve the compound inequality: x is less than 3 or x is greater than or equal to 5.

Accepted Answer

x < 3 or x > 5

Answer

x < 3 and x ≥ 5

Answer

x > 3 and x ≤ 5

Answer

x > 3 or x < 5

Question 2

Identify the inequality equivalent to |x| is less than or equal to 2.

Accepted Answer

-2 ≤ x ≤ 2

Answer

x ≤ -2 or x ≥ 2

Answer

-2 < x < 2

Answer

x < -2 or x > 2

Question 3

Select the inequality represented by the set notation: {x | x is less than or equal to 0}.

Accepted Answer

x ≤ 0

Answer

x > 0

Answer

x < 0

Answer

x ≥ 0

Question 4

Write the set-builder notation that represents the solution set of x > 5

Accepted Answer

{x | x > 5}

Answer

{x | x < 5}

Answer

{x | x ≥ 5}

Answer

{x | x ≤ 5}

Question 5

Solve the inequality: 2x - 5 is greater than 7

Accepted Answer

x is greater than 6

Answer

x is less than 6

Answer

x is greater than 12

Answer

x is less than 12

Question 6

Solve the inequality: 3x + 5 > 14

Accepted Answer

x > 3

Answer

x < 3

Answer

x > 9

Answer

x < 9

Question 7

Solve the inequality: 
|x - 2| < 5

Accepted Answer

-3 < x < 7

Answer

x < -3 or x > 7

Answer

-7 < x < 3

Answer

x > -7 or x < 3

Question 8

Which of the following inequalities represents the statement: “The sum of x and y is greater than 10”?

Accepted Answer

x + y > 10

Answer

x + y < 10

Answer

x - y > 10

Answer

x - y < 10

Question 9

Solve the inequality: x + 5 > 12

Accepted Answer

x > 7

Answer

x < 7

Answer

x < 5

Answer

x > 5

Question 10

Solve the inequality graphically: x + 2y is less than 6.

Accepted Answer

The shaded region below the line y = -0.5x + 3.

Answer

The shaded region above the line y = -0.5x + 3.

Answer

The shaded region below the line y = 0.5x - 3.

Answer

The shaded region above the line y = 0.5x - 3.

Question 11

Solve the compound inequality: -3 is less than x + 2 and x + 2 is less than 5.

Accepted Answer

-3 is less than x and x is less than 5

Answer

-5 is less than x and x is less than 3

Answer

-1 is less than x and x is less than 5

Answer

-1 is less than x and x is less than 3

Question 12

Which inequality has the solution set (negative infinity, 2) union (3, infinity)?

Accepted Answer

x is less than 2 or x is greater than 3

Answer

x is greater than 2 or x is less than 3

Answer

x is less than 2 or x is greater than -3

Answer

x is greater than 2 or x is less than -3

Question 13

Which of the following is NOT a solution to the inequality: x^2 + 2x - 3 is greater than 0?

Accepted Answer

x equals -3

Answer

x equals 1

Answer

x equals 2

Answer

x equals -1

Question 14

Which of the following is equivalent to the inequality: 3(x - 2) > 6?

Accepted Answer

x > 4

Answer

x < 4

Answer

x > 2

Answer

x < 2

Question 15

Find the interval notation for the inequality: -2 < x ≤ 5

Accepted Answer

(-2, 5]

Answer

[-2, 5]

Answer

(-∞, 5]

Answer

[-2, ∞)

Question 16

Solve the inequality: |2x - 3| ≤ 5

Accepted Answer

-1 ≤ x ≤ 4

Answer

0 ≤ x ≤ 4

Answer

-4 ≤ x ≤ 1

Answer

x ≤ -1 or x ≥ 4

Question 17

Which of the following inequalities is true for all values of x?

Accepted Answer

x ≥ 0

Answer

x > 0

Answer

x < 0

Answer

x ≤ 0

Question 18

Solve the inequality: -3(2x - 1) > 6

Accepted Answer

x < -1

Answer

x > -1

Answer

x < 1

Answer

x > 1

Question 19

Find the solution set of the inequality: x^2 - 4 > 0

Accepted Answer

x < -2 or x > 2

Answer

x > -2 and x < 2

Answer

x < -2 and x > 2

Answer

x > -2 or x < 2

Question 20

What is the solution set of the inequality x - 5 < 2?

Accepted Answer

x < 7

Answer

x > 7

Answer

x < -7

Answer

x > -7

Question 21

Solve the inequality: 2(x + 3) - 5 > 11

Accepted Answer

x > 5

Answer

x < 5

Answer

x > -5

Answer

x < -5

Question 22

Which inequality is equivalent to x - 4 > 6?

Accepted Answer

x > 10

Answer

x > 2

Answer

x < 10

Answer

x < 2

Question 23

Solve the system of inequalities: x + 2 > 5 and x - 3 < 1

Accepted Answer

x > 3 and x < 4

Answer

x < 4

Answer

x > 4

Answer

No solution

Question 24

Solve the inequality: (x-2)(x+3) < 0

Accepted Answer

x < -3 or x > 2

Answer

x > -3 and x < 2

Answer

x < -3 or x < 2

Answer

x > -3 and x > 2

Question 25

Graph the inequality: 2x-1 ≥ 5

Accepted Answer

A line with a slope of 2, solid above x = 3

Answer

A line with a slope of 2, dashed above x = 3

Answer

A line with a slope of 1/2, dashed above x = 3

Answer

A line with a slope of 1/2, solid above x = 3

Question 26

Express the inequality 3x - 5 < 0 using interval notation

Accepted Answer

(-∞ , 5/3)

Answer

(-∞ , 3/5)

Answer

(5/3 , ∞)

Answer

(3/5 , ∞)

Question 27

Which of the following is NOT a solution to the inequality x - 3 ≤ 5?

Accepted Answer

x = 9

Answer

x = 1

Answer

x = 7

Answer

x = -1

Question 28

Solve the inequality: 2x + 3 is less than 11

Accepted Answer

x is less than 4

Answer

x is greater than 4

Answer

x is less than 2

Answer

x is greater than 2

Question 29

Which of the following represents the solution set to the inequality: 3x - 5 is greater than or equal to 10?

Accepted Answer

{x|x is greater than or equal to 5}

Answer

{x|x is greater than 5}

Answer

{x|x is less than 5}

Answer

{x|x is less than or equal to 5}

Question 30

Which of the following inequalities is equivalent to 2x - 3 is greater than 5?

Accepted Answer

x is greater than 4

Answer

x is less than 4

Answer

x is less than 2

Answer

x is greater than 2

Question 31

Solve the inequality: negative 2 times (x plus 3) is less than -6

Accepted Answer

x is less than -3

Answer

x is greater than -3

Answer

x is greater than 3

Answer

x is less than 3

Question 32

Solve the inequality: 2x + 5 > 13

Accepted Answer

x > 4

Answer

x < -4

Answer

x < 4

Answer

x > -4

Question 33

Solve the following inequality: x - 5 > 10

Accepted Answer

x > 15

Answer

x > -5

Answer

x < 15

Answer

x < -5

Question 34

Which of the following represents the solution set of the inequality x > 3 in set-builder notation?

Accepted Answer

(3, ∞)

Answer

(-∞, 3]

Answer

[3, ∞)

Answer

(∞, 3)

Question 35

Which of the following is equivalent to the inequality: x² - 4 < 0?

Accepted Answer

-2 < x < 2

Answer

x > -2 or x < 2

Answer

-2 ≤ x ≤ 2

Answer

x < -2 or x > 2

Question 36

Solve the inequality: -3x ≤ 9

Accepted Answer

x ≥ -3

Answer

x ≤ -3

Answer

x < -3

Answer

x > -3

Question 37

Translate the following statement into a mathematical inequality: "The sum of twice a number and 3 is less than 15."

Accepted Answer

2x + 3 < 15

Answer

2x - 3 < 15

Answer

2x - 3 ≥ 15

Answer

2x + 3 > 15

Question 38

Check if the following number is a solution to the inequality 4x - 1 > 7: x = 3

Accepted Answer

Yes

Answer

No

Question 39

Express the solution set for the inequality 3x + 2 ≤ 8 using set-builder notation.

Accepted Answer

{x | x ≤ 2}

Answer

{x | x < 2}

Answer

{x | x ≥ 2}

Question 40

When graphing the inequality x > 5 on a number line, what symbol should be used at the point x = 5?

Accepted Answer

Open circle

Answer

Triangle

Answer

Closed circle

Answer

Square

Question 41

Solve the inequality: 2x + 5 > 11

Accepted Answer

x > 3

Answer

x < 3

Answer

x ≥ 3

Answer

x ≤ 3

Question 42

Translate the following phrase into a mathematical inequality: "A number is at least 5."

Accepted Answer

x ≥ 5

Answer

x ≤ 5

Answer

x > 5

Answer

x < 5

Question 43

Solve the inequality: 4x - 7 < 5

Accepted Answer

x < 3

Answer

x > 3

Answer

x ≤ 3

Answer

x ≥ 3

Question 44

Express the solution to the inequality -2(x + 3) > 8 in set-builder notation.

Accepted Answer

{x | x < -7}

Answer

{x | x ≤ -7}

Answer

{x | x > -7}

Answer

{x | x ≥ -7}

Question 45

If 3x + 2 ≤ 14, which of the following is a possible value for x?

Accepted Answer

x = 4

Answer

x = 7

Answer

x = 6

Answer

x = 5

Question 46

The cost of renting a car is $25 per day plus $0.15 per mile driven. If you have a budget of $100, which inequality represents the maximum number of miles (m) you can drive?

Accepted Answer

0.15m + 25 ≤ 100

Answer

0.15m - 25 > 100

Answer

0.15m - 25 ≤ 100

Answer

0.15m + 25 > 100

Question 47

The length of a rectangle is 3 more than twice its width (w). If the perimeter of the rectangle is at most 30 units, which inequality represents the possible values of the width?

Accepted Answer

2(2w + 3) + 2w ≤ 30

Answer

2(2w + 3) - 2w > 30

Answer

2(2w + 3) - 2w ≤ 30

Answer

2(2w + 3) + 2w > 30