Quadratic inequalities are an essential part of algebra, involving expressions where a quadratic polynomial is set to be greater than or less than a certain value. Practicing quadratic inequalities helps in understanding the range of values that satisfy these inequalities, which is crucial for solving real-world problems and preparing for exams.

In simple terms, a quadratic inequality looks like ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, and ax² + bx + c ≤ 0. The solutions to these inequalities are the ranges of x that make the inequality true. To find these solutions, we often factorize the quadratic expression, find its roots, and then determine the intervals where the expression meets the inequality condition.

What is Inequality?

An inequality is a mathematical statement that shows the relationship between two values when they are not equal. Inequalities express that one quantity is larger or smaller than another and use symbols like >, <, ≥, and ≤.

Using this symbols many types of inequalities can be created such as:

• Linear Inequality
• Quadratic Inequality
• etc.

What is Quadratic Inequality?

A quadratic inequality is an inequality that involves a quadratic expression, which is a polynomial of degree two. These inequalities take the form:

• ax² + bx + c > 0
• ax² + bx + c < 0
• ax² + bx + c ≥ 0, and
• ax² + bx + c ≤ 0

Where a, b, and c are constants, and a ≠ 0.

Examples of Quadratic Inequality

Examples of quadratic inequalities are:

• Strict Inequality:
  • x² − 3x + 2 > 0
  • 2x² + 4x − 6 < 0
• Non-strict Inequality:
  • x² − 5x + 6 ≥ 0
  • 3x² − 2x + 1 ≤ 0

Solved Problems on Quadratic Inequalities

Problem 1: Solve x² – 3x – 4 > 0.

Solution:

Given: x² – 3x – 4 > 0

Factorize the Quadratic Expression:

x² – 3x – 4 = (x – 4)(x + 1)

The roots are x = 4 and x = -1.

The critical points divide the number line into three intervals: (-∞, -1), (-1, 4), and (4, ∞).

For x ∈ (-∞, -1), choose x = -2:

(-2 – 4)(-2 + 1) = (-6)(-1) = 6, (> 0)

For x ∈ (-1, 4), choose x = 0:

(0 – 4)(0 + 1) = (-4)(1) = -4, (< 0)

For x ∈ (4, ∞), choose x = 5:

(5 – 4)(5 + 1) = (1)(6) = 6, (> 0)

The inequality x² – 3x – 4 > 0 is satisfied for x ∈ (-∞, -1) ∪ (4, ∞).

Problem 2: Solve 2x² + 3x – 2 ≤ 0.

Solution:

Given: 2x² + 3x – 2 ≤ 0

2x² + 3x – 2 = (2x – 1)(x + 2)

The roots are x = 1/2 and x = -2.

The critical points divide the number line into three intervals: (-∞, -2), (-2, 1/2), and (1/2, ∞).

For x ∈ (-∞, -2), choose x = -3:

(2(-3) – 1)((-3) + 2) = (-7)(-1) = 7, (> 0)

For x ∈ (-2, 1/2), choose x = 0:

(2(0) – 1)(0 + 2) = (-1)(2) = -2, (< 0)

For x ∈ (1/2, ∞), choose x = 1:

(2(1) – 1)(1 + 2) = (1)(3) = 3, (> 0)

The inequality 2x² + 3x – 2 ≤ 0 is satisfied for x ∈ [-2, 1/2].

Problem 3: Solve x² – 4x + 4 ≥ 0

Solution:

Given: x² – 4x + 4 ≥ 0

x² – 4x + 4 = (x – 2)²

The root is x = 2.

The critical point divides the number line into two intervals: (-∞, 2) and (2, ∞).

For x ∈ (-∞, 2), choose x = 1:

(1 – 2)² = 1, (> 0)

For x ∈ (2, ∞), choose x = 3:

(3 – 2)² = 1, (> 0)

At x = 2:

(2 – 2)² = 0, (= 0)

The inequality x² – 4x + 4 ≥ 0 is satisfied for all x ∈ R.

Problem 4: Solve x² + x – 6 < 0

Solution:

Given: x² + x – 6 < 0

x² + x – 6 = (x – 2)(x + 3)

The roots are x = 2 and x = -3.

The critical points divide the number line into three intervals: (-∞, -3), (-3, 2), and (2, ∞).

For x ∈ (-∞, -3), choose x = -4:

(-4 – 2)(-4 + 3) = (-6)(-1) = 6, (> 0)

For x ∈ (-3, 2), choose x = 0:

(0 – 2)(0 + 3) = (-2)(3) = -6, (< 0)

For x ∈ (2, ∞), choose x = 3:

(3 – 2)(3 + 3) = (1)(6) = 6, (> 0)

The inequality x² + x – 6 < 0 is satisfied for x ∈ (-3, 2).

Problem 5: Solve 3x² – 5x + 2 ≥ 0

Solution:

Given: 3x² – 5x + 2 ≥ 0

3x² – 5x + 2 = (3x – 2)(x – 1)

The roots are x = 2/3 and x = 1.

The critical points divide the number line into three intervals: (-∞, 2/3), (2/3, 1), and (1, ∞).

For x ∈ (-∞, 2/3), choose x = 0:

(3(0) – 2)(0 – 1) = (-2)(-1) = 2, (> 0)

For x ∈ (2/3, 1), choose x = 0.8:

(3(0.8) – 2)(0.8 – 1) = (0.4)(-0.2) = -0.08, (< 0)

For x ∈ (1, ∞), choose x = 2:

(3(2) – 2)(2 – 1) = (4)(1) = 4, (> 0)

The inequality 3x² – 5x + 2 ≥ 0 is satisfied for x ∈ (-∞, 2/3] ∪ [1, ∞).

Problem 6: Solve x² + 4x – 5 ≤ 0

Solution:

Given: x² + 4x – 5 ≤ 0

x² + 4x – 5 = (x + 5)(x – 1)

The roots are x = -5 and x = 1.

The critical points divide the number line into three intervals: (-∞, -5), (-5, 1), and (1, ∞).

For x ∈ (-∞, -5), choose x = -6:

(-6 + 5)(-6 – 1) = (-1)(-7) = 7, (> 0)

For x ∈ (-5, 1), choose x = 0:

(0 + 5)(0 – 1) = (5)(-1) = -5, (< 0)

For x ∈ (1, ∞), choose x = 2:

(2 + 5)(2 – 1) = (7)(1) = 7, (> 0)

The inequality x² + 4x – 5 ≤ 0 is satisfied for x ∈ [-5, 1].

Problem 7: Solve 2x² + x – 3 > 0

Solution:

Given: 2x² + x – 3 > 0

2x² + x – 3 = (2x – 3)(x + 1)

The roots are x = 3/2 and x = -1.

The critical points divide the number line into three intervals: (-∞, -1), (-1, 3/2), and (3/2, ∞).

For x ∈ (-∞, -1), choose x = -2:

(2(-2) – 3)((-2) + 1) = (-7)(-1) = 7, (> 0)

For x ∈ (-1, 3/2), choose x = 0:

(2(0) – 3)(0 + 1) = (-3)(1) = -3, (< 0)

For x ∈ (3/2, ∞), choose x = 2:

(2(2) – 3)(2 + 1) = (1)(3) = 3, (> 0)

The inequality 2x² + x – 3 > 0 is satisfied for x ∈ (-∞, -1) ∪ (3/2, ∞).

Problem 8: Solve x² + 2x – 8 ≤ 0

Solution:

Given: x² + 2x – 8 ≤ 0

x² + 2x – 8 = (x + 4)(x – 2)

The roots are x = -4 and x = 2.

The critical points divide the number line into three intervals: (-∞, -4), (-4, 2), and (2, ∞).

For x ∈ (-∞, -4), choose x = -5:

(-5 + 4)(-5 – 2) = (-1)(-7) = 7, (> 0)

For x ∈ (-4, 2), choose x = 0:

(0 + 4)(0 – 2) = (4)(-2) = -8, (< 0)

For x ∈ (2, ∞), choose x = 3:

(3 + 4)(3 – 2) = (7)(1) = 7, (> 0)

The inequality x² + 2x – 8 ≤ 0 is satisfied for x ∈ [-4, 2].

Problem 9: Solve 4x² – 4x + 1 ≥ 0

Solution:

Given: 4x² – 4x + 1 ≥ 0

4x² – 4x + 1 = (2x – 1)²

The root is x = 1/2.

The critical point divides the number line into two intervals: (-∞, 1/2) and (1/2, ∞).

For x ∈ (-∞, 1/2), choose x = 0:

(2(0) – 1)² = (-1)^2 = 1, (> 0)

For x ∈ (1/2, ∞), choose x = 1:

(2(1) – 1)² = (1)² = 1, (> 0)

At x = 1/2:

(2(1/2) – 1)² = 0² = 0, (= 0)

The inequality 4x² – 4x + 1 ≥ 0 is satisfied for all x ∈R.

Problem 10: Solve x² – 2x > 3

Solution:

Given: x² – 2x – 3 > 0

x² – 2x – 3 = (x – 3)(x + 1)

The roots are x = 3 and x = -1.

The critical points divide the number line into three intervals: (-∞, -1), (-1, 3), and (3, ∞).

For x ∈ (-∞, -1), choose x = -2:

(-2 – 3)(-2 + 1) = (-5)(-1) = 5, (> 0)

For x ∈ (-1, 3), choose x = 0:

(0 – 3)(0 + 1) = (-3)(1) = -3, (< 0)

For x ∈ (3, ∞), choose x = 4:

(4 – 3)(4 + 1) = (1)(5) = 5, (> 0)

The inequality x² – 2x > 3 is satisfied for x ∈ (-∞, -1) ∪ (3, ∞).

Practice Problems on Quadratic Inequalities

Problem 1: Solve the inequality x² – 6x + 8 < 0.

Problem 2:Determine the solution set for 2x² – 5x + 2 ≥ 0.

Problem 3: Find the values of x that satisfy x² + 2x – 15 > 0.

Problem 4: Solve for x in the inequality 3x² + x – 4 ≤ 0.

Problem 5: Determine the solution set for 4x² – 12x + 9 ≥ 0.

Problem 6: Find the values of x that satisfy x² – 9x + 14 < 0.

Problem 7: Solve the inequality x² + 4x + 4 > 0.

Problem 8: Determine the solution set for 5x² – 3x – 2 ≤ 0.

Problem 9: Solve for x in the inequality 2x² – 7x + 3 > 0.

Problem 10: Find the values of x that satisfy x² – 2x – 8 ≤ 0.

Read More,

• Inequalities
• Inequalities Practice Questions with Solution
• Linear Inequalities
• Graphical Solution of Linear Inequalities in Two Variables
• Solving Linear Inequalities Word Problems
• Quadratic Inequalities

FAQs: Quadratic Inequalities

What is a quadratic inequality?

A quadratic inequality is an inequality that involves a quadratic expression, typically in the form ax² + bx + c, where a ≠ 0. It can be expressed using inequality signs such as >, <, ≥, or ≤.

How do you solve a quadratic inequality?

Solving a quadratic inequality involves:

• Factoring the quadratic expression (if possible).
• Finding the roots of the corresponding quadratic equation.
• Using the roots to divide the number line into intervals.
• Testing each interval to determine where the inequality holds true.
• Writing the solution set based on the intervals that satisfy the inequality.

What is the difference between a quadratic equation and a quadratic inequality?

A quadratic equation is an equation of the form ax² + bx + c and has specific solutions (roots). A quadratic inequality, on the other hand, uses inequality signs and describes a range of values for which the inequality is true.