Inequalities: Master the Art of Solving Inequalities

Question 1

Solve for x: |x - 2| < 5. The solution is the set of all x that satisfy the inequality.

Accepted Answer

-3 < x < 7

Answer

-7 < x < 3

Answer

x < -3 or x > 7

Answer

x < -7 or x > 3

Question 2

Which of the following is a compound inequality? An inequality that contains two or more inequalities joined by the word 'and' or 'or'.

Accepted Answer

x ≤ 2 or x ≥ 4

Answer

x > 5

Answer

x + y ≥ 10

Answer

y - 3 < 0

Question 3

Solve the inequality 3x + 2 > 7.

Accepted Answer

x > 1

Answer

x < 1

Answer

x < 1.5

Answer

x > 1.5

Question 4

Which of the following is NOT a representation of an inequality?

Accepted Answer

x = 5

Answer

2x + 5 > 11

Answer

y - 3 ≤ 7

Answer

3a ≥ 15

Question 5

Solve for x: 2(x - 3) > 5

Accepted Answer

x > 4

Answer

x > 2

Answer

x < 2

Answer

x < 4

Question 6

Solve the system of inequalities: x + y > 5, y ≤ 2

Accepted Answer

x > 5 and y ≤ 2

Answer

x > 5

Answer

y > 5

Answer

y > 5 and x ≤ 2

Question 7

What is the opposite of the inequality: x + 5 is greater than 3?

Accepted Answer

x + 5 is less than 3

Answer

x + 5 is less than or equal to 3

Answer

x - 5 is greater than 3

Answer

x + 5 is equal to 3

Question 8

Translate the following word problem into an inequality: "Three times a number minus two is less than or equal to the number plus four."

Accepted Answer

3x - 2 ≤ x + 4

Answer

3x - 2 > x + 4

Answer

3(x - 2) ≤ x + 4

Answer

3(x - 2) > x + 4

Question 9

Solve the inequality: x + 3 > 7. What is the value of x?

Accepted Answer

x > 4

Answer

x < 4

Answer

x > 10

Answer

x < 10

Question 10

Which graph represents the inequality y ≤ 2x + 1?

Accepted Answer

A line with a slope of 2, a y-intercept of 1, and shading below the line.

Answer

A line with a slope of 2, a y-intercept of -1, and shading above the line.

Answer

A line with a slope of -2, a y-intercept of 1, and shading below the line.

Answer

A line with a slope of -2, a y-intercept of -1, and shading above the line.

Question 11

Which system of inequalities has no solution?

Accepted Answer

x < 2 and x > 5

Answer

x > 2 and x < 5

Answer

x > 2 or x < 5

Answer

x > 2 and x ≤ 5

Question 12

Which symbol is NOT used to represent an inequality?

Accepted Answer

=

Answer

>

Answer

<

Answer

≥

Question 13

Which inequality represents the statement: "A number x is at most negative five"?

Accepted Answer

x ≤ -5

Answer

x ≥ -5

Answer

x > -5

Answer

x < -5

Question 14

Which of the following is equivalent to the inequality: |x - 3| < 2?

Accepted Answer

-1 < x < 5

Answer

-5 < x < 1

Answer

x > -1 or x < 5

Answer

x < -5 or x > 1

Question 15

Which inequality is considered a compound inequality?

Accepted Answer

x ≤ -2 and x ≥ 3

Answer

x > 5

Answer

y - 1 < 0

Answer

2x + 1 ≥ 5

Question 16

Which of the following is an inequality?

Accepted Answer

x > 5

Answer

x = 5

Answer

x + y = 10

Answer

x - y = 0

Question 17

Graph the inequality: y ≤ 2x + 1 on a coordinate plane

Accepted Answer

A line with slope 2 and y-intercept 1, shaded below the line

Answer

A line with slope 2 and y-intercept 1, shaded above the line

Answer

A line with slope -2 and y-intercept -1, shaded above the line

Answer

A line with slope -2 and y-intercept -1, shaded below the line

Question 18

Which of the following inequalities represents the statement "The sum of x and y is less than 10"?

Accepted Answer

x + y < 10

Answer

x + y > 10

Answer

x - y < 10

Answer

x - y > 10

Question 19

Solve the inequality: |x - 2| > 3

Accepted Answer

x < -1 or x > 5

Answer

x < 1 or x > 5

Answer

x < -5 or x > 1

Answer

x < -3 or x > 3

Question 20

Which of the following is the solution set of the inequality: x^2 - 4 < 0?

Accepted Answer

(-2, 2)

Answer

[-2, 2]

Answer

(-∞, -2) ∪ (2, ∞)

Answer

[2, ∞)

Question 21

Which of the following inequalities is equivalent to x > 5?

Accepted Answer

5 < x

Answer

-x < -5

Answer

x < -5

Answer

5 > x

Question 22

Identify the inequality that represents "x is greater than or equal to 5."

Accepted Answer

x ≥ 5

Answer

x < 5

Answer

x ≤ 5

Answer

x > 5

Question 23

Solve the compound inequality: 1 < x ≤ 4

Accepted Answer

1 ≤ x ≤ 4

Answer

x < 1 or x > 4

Answer

x ≥ 1 and x ≤ 4

Answer

1 < x or x < 4

Question 24

Find the range of values for x that satisfy the system of inequalities: 3x + 5 > 14 and x - 2 < 5.

Accepted Answer

0 < x < 7

Answer

x > 3 or x < 7

Answer

x < 3 or x > 7

Answer

There is no solution.

Question 25

Identify the inequality that expresses that the value of x is greater than 2.

Accepted Answer

x > 2

Answer

x ≥ 2

Answer

x < 2

Question 26

Which of the following statements is true for the inequality |2x - 5| > 3?

Accepted Answer

x < 1 or x > 4

Answer

x < -1 or x > -4

Answer

x > 1 or x < 4

Answer

x > -1 or x < -4

Question 27

Solve the compound inequality: -5 < 3x + 2 ≤ 11

Accepted Answer

-1 < x ≤ 3

Answer

x > -1 or x < 3

Answer

-2 < x < 2

Answer

x < -1 or x > 3

Question 28

Which of the following inequalities is equivalent to x² - 9 > 0?

Accepted Answer

(x - 3)(x + 3) > 0

Answer

(x - 3)(x + 3) < 0

Answer

(x - 9)(x + 9) > 0

Answer

(x - 9)(x + 9) < 0

Question 29

Solve the system of inequalities:
    2x - y < 6
    x + 2y > 8

Accepted Answer

y < 2x + 6 and y > 4 - (x/2)

Answer

y > 2x - 6 and y < 4 - (x/2)

Answer

y > 2x - 6 and y > 4 - (x/2)

Answer

y < 2x + 6 and y < 4 - (x/2)

Question 30

Translate the following statement into an inequality: 'The number x is at least 7'

Accepted Answer

x ≥ 7

Answer

x > 7

Answer

x < 7

Answer

x ≤ 7

Question 31

Find the domain of the inequality: x / (x - 2) > 0

Accepted Answer

(−∞, 2) ∪ (2, ∞)

Answer

(−∞, 2]

Answer

All real numbers

Answer

[2, ∞)

Question 32

Which inequality is true for all values of x?

Accepted Answer

x² ≥ 0

Answer

x² > 0

Answer

x² < 0

Answer

x² ≤ 0

Question 33

What inequality represents the phrase 'The product of a number *x* and 5 is at least 20'?

Accepted Answer

5x ≥ 20

Answer

5x < 20

Answer

5x ≤ 20

Answer

5x > 20

Question 34

Which inequality matches the graph below? (The shaded area shows the solutions)

Accepted Answer

y > -x + 2

Answer

y ≤ -x + 2

Answer

y < -x + 2

Answer

y ≥ -x + 2

Question 35

A rectangular garden needs to have a perimeter less than 50 meters.  The length is 3 meters longer than the width. Which inequality shows the possible widths (*w*) of the garden?

Accepted Answer

2w + 2(w + 3) < 50

Answer

w(w + 3) < 50

Answer

w + 3w < 50

Answer

2w + 3 < 50

Question 36

What are the solutions to the inequality |x + 2| < 5?

Accepted Answer

-7 < x < 3

Answer

x < 3 or x > -7

Answer

-3 < x < 3

Answer

x < -7 or x > 3

Question 37

Which value is *NOT* a solution to the compound inequality -4 ≤ 2x + 6 < 10?

Accepted Answer

x = -5

Answer

x = 0

Answer

x = -2

Answer

x = 1

Question 38

Renting a bicycle costs $10 plus $5 per hour. Sarah has $30 to spend. What inequality shows how many hours (*h*) she can rent the bike?

Accepted Answer

10 + 5h ≤ 30

Answer

10 + 5h ≥ 30

Answer

10h + 5 ≤ 30

Question 39

Which point is a solution to this system of inequalities: y > x - 2 and y ≤ -2x + 3?

Accepted Answer

(0, 1)

Answer

(2, 2)

Answer

(-1, 3)

Answer

(1, -1)

Question 40

Solve the inequality: 2x + 5 > 15

Accepted Answer

x > 5

Answer

x > -5

Answer

x < -5

Answer

x < 5

Question 41

What is the solution to the inequality: |x - 3| ≤ 2?

Accepted Answer

1 ≤ x ≤ 5

Answer

3 ≤ x ≤ 5

Answer

1 ≤ x ≤ 3

Answer

-1 ≤ x ≤ 5

Question 42

What is the solution to the inequality: 3x - 2 > 5?

Accepted Answer

x > 2.33

Answer

x < 2.33

Answer

x > -2.33

Answer

x < -2.33

Question 43

Which of the following inequalities represents the condition that the sum of two numbers is less than 10?

Accepted Answer

x + y < 10

Answer

x + y ≤ 10

Answer

x + y ≥ 10

Answer

x + y > 10

Question 44

Which of the following inequalities is equivalent to x - 3 ≤ 5?

Accepted Answer

x ≤ 8

Answer

x ≤ 5

Answer

x ≥ 8

Answer

x ≥ 5

Question 45

Which of the following inequalities represents the condition that the difference between two numbers is greater than or equal to 5?

Accepted Answer

x - y ≥ 5

Answer

x - y ≤ 5

Answer

x - y < 5

Answer

x - y > 5