Solving Inequalities: Master Linear and Quadratic Inequalities with Ease

Question 1

Which inequality is represented by the shaded region below the x-axis in the graph?

Accepted Answer

y < 0

Answer

y > 0

Answer

x < 0

Answer

x > 0

Question 2

A company needs to produce at least 500 units of a product to make a profit. Write the inequality that represents this situation, where x is the number of units produced.

Accepted Answer

x ≥ 500

Answer

x ≤ 500

Answer

x > 500

Answer

x < 500

Question 3

Graph the inequality: x² - 4 is greater than 0

Accepted Answer

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

Answer

x is greater than -2 and x is less than 2

Answer

x is less than -4 or x is greater than 0

Answer

x is greater than -4 and x is less than 0

Question 4

Which of the following is the solution to the inequality: |x - 3| is greater than 2

Accepted Answer

x is less than 1 or x is greater than 5

Answer

x is greater than 1 and x is less than 5

Answer

x is less than -1 or x is greater than -5

Answer

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

Question 5

Find the solution set for the inequality 3x - 5 > 10.

Accepted Answer

x > 5

Answer

x < 5

Answer

x < -5

Answer

x > -5

Question 6

Solve the quadratic inequality x² - 4x + 3 < 0.

Accepted Answer

x < 1 or x > 3

Answer

1 < x < 3

Answer

x > 1 or x < 3

Answer

x < 1 and x > 3

Question 7

Find the solution set for the inequality 2(x + 1) - 3(x - 2) ≤ 5.

Accepted Answer

x ≤ 1

Answer

x ≥ 1

Answer

x ≤ -1

Answer

x ≥ -1

Question 8

Solve the inequality |x - 2| ≤ 3.

Accepted Answer

-1 ≤ x ≤ 5

Answer

-5 ≤ x ≤ 1

Answer

1 ≤ x ≤ 5

Answer

x ≤ 1 or x ≥ 5

Question 9

Which graph represents the inequality 2x - 5 > 7?

Accepted Answer

x > 6

Answer

x < 6

Answer

x > -1

Answer

x < -1

Question 10

Solve the linear inequality: 2x - 5 > 7

Accepted Answer

x > 6

Answer

x < 6

Answer

x > -6

Answer

x < -6

Question 11

Which graphical representation corresponds to the quadratic inequality: x^2 - 4 < 0?

Accepted Answer

A parabola opening downwards, with a solid line at x = 2 and x = -2

Answer

A parabola opening upwards, with a dashed line at x = 2 and x = -2

Answer

A parabola opening upwards, with a solid line at x = 2 and x = -2

Answer

A parabola opening downwards, with a dashed line at x = 2 and x = -2

Question 12

Solve the absolute value inequality: |x - 2| > 3

Accepted Answer

x < -1 or x > 5

Answer

-1 < x < 5

Answer

-5 < x < 1

Answer

x < -5 or x > 1

Question 13

Solve the linear inequality: 2x - 5 ≤ 15

Accepted Answer

x ≤ 10

Answer

x < 10

Answer

x > 10

Answer

x ≥ 10

Question 14

Which inequality is equivalent to 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 15

In engineering, inequalities are used to determine which of the following?

Accepted Answer

The maximum load a bridge can support safely.

Answer

The length of a side in a right-angled triangle.

Answer

The precise area of a circular component.

Answer

The exact volume of a spherical tank.

Question 16

Which inequality is always true, no matter the value of x?

Accepted Answer

x² ≥ 0

Answer

x² - 4 < 0

Answer

x² - 4 ≤ 0

Answer

x² - 4 > 0

Question 17

Solve the inequality: 2x - 5 > 7

Accepted Answer

x > 6

Answer

x < 6

Answer

x > -6

Answer

x < -6

Question 18

The inequality 2x² - 5x + 2 ≤ 0 represents a parabola.  What are the coordinates of this parabola's vertex?

Accepted Answer

(5/4, -3/8)

Answer

(-5/4, -3/8)

Answer

(4/5, -8/3)

Answer

(3/8, -5/4)

Question 19

Which of the following inequalities can be used to model the statement "The sum of two numbers x and y is at least 10"?

Accepted Answer

x + y ≥ 10

Answer

x + y < 10

Answer

x + y ≤ 10

Answer

x + y > 10

Question 20

Which of the following inequalities represents 'x is greater than -3 and less than or equal to 5'?

Accepted Answer

-3 < x ≤ 5

Answer

-3 > x ≥ 5

Answer

-3 ≤ x < 5

Question 21

Solve for x: 2x + 5 < 11

Accepted Answer

x < 3

Answer

x > 3

Answer

x ≥ 3

Answer

x ≤ 3

Question 22

If 3x - 7 ≥ 2, what is the smallest possible integer value of x?

Accepted Answer

3

Answer

5

Answer

2

Answer

4

Question 23

Which number line represents the solution to the inequality -2(x + 1) ≤ 4?

Accepted Answer

Image showing a number line with a closed circle on -3 and an arrow pointing to the right.

Answer

Image showing a number line with a closed circle on 3 and an arrow pointing to the left.

Answer

Image showing a number line with an open circle on -3 and an arrow pointing to the left.

Question 24

Solve the inequality: x² - 5x + 6 > 0

Accepted Answer

x < 2 or x > 3

Answer

2 < x < 3

Answer

x ≤ 2 or x ≥ 3

Answer

2 ≤ x ≤ 3

Question 25

The area of a rectangle is less than 24 square units. The length of the rectangle is (x + 2) units and the width is (x - 1) units. Which inequality represents this situation?

Accepted Answer

(x + 2)(x - 1) < 24

Answer

(x + 2)(x - 1) > 24

Answer

(x + 2) + (x - 1) < 24

Question 26

For what values of x is the expression √(x - 5) defined in the real number system?

Accepted Answer

x ≥ 5

Answer

x > 5

Answer

x < 5

Answer

x ≤ 5

Question 27

Which of the following points lies within the solution region of the inequality system: y > x - 2 and y ≤ -x + 3?

Accepted Answer

(1, 1)

Answer

(-1, -2)

Answer

(0, 4)

Answer

(3, 2)

Question 28

If -2 < x < 5 and 3 < y < 7, what is the maximum possible value of x - y?

Accepted Answer

-1

Answer

2

Answer

12

Answer

-6

Question 29

The profit, P, made from selling x units of a product is modeled by P = -0.5x² + 20x - 50. For what values of x is the profit positive (P > 0)?

Accepted Answer

5 < x < 15

Answer

x < 0 or x > 40

Answer

0 < x < 40

Answer

x < 5 or x > 15