Linear inequalities are fundamental concepts in algebra that help us understand and solve a wide range of real-world problems. These inequalities express the relationship between two expressions using inequality symbols like <, ≤, >, and ≥. Learning to solve linear inequalities involves understanding how to manipulate these expressions to isolate the variable and determine the range of possible solutions.

In this article, we will explore various practice questions on linear inequalities to enhance your understanding and problem-solving skills. We will cover single-variable inequalities, systems of linear inequalities, and graphical representations of these solutions. Each section includes step-by-step explanations to help you grasp the methods used in solving different types of inequalities.

What are Linear Inequalities?

Linear inequalities are mathematical expressions involving linear functions that use inequality symbols (such as <, ≤, >, or ≥) instead of an equality sign. These inequalities describe a relationship where one side of the expression is not strictly equal to the other but is either less than, greater than, or equal to (depending on the inequality used).

Examples of Linear Inequalities

• One-Step Linear Inequality: x − 7 > 10
• Two-Step Linear Inequality: 5x + 4 ≤ 19
• Linear Inequality with Negative Coefficient: 20 − 3x < 8
• Compound Inequality: −3 < 4x + 1 ≤ 17

Solved Problems on Linear Inequalities

Problem 1: Solve the inequality: x – 5 ≤ 3.

Solution:

Given: x – 5 ≤ 3

Add 5 to both sides to isolate x:

x – 5 + 5 ≤ 3 + 5

⇒ x ≤ 8

Thus, x ∈ (-∞, 8]

Problem 2: Solve the inequality: 4x – 7 > 9

Solution:

Given: 4x – 7 > 9

Add 7 to both sides:

4x – 7 + 7 > 9 + 7

⇒ 4x > 16

Divide both sides by 4:

x > 4

Thus, x ∈ (4, ∞)

Problem 3: Solve the inequality: -2x + 6 ≥ 8

Solution:

Given: -2x + 6 ≥ 8

Subtract 6 from both sides:

-2x + 6 – 6 ≥ 8 – 6

⇒ -2x ≥ 2

Divide both sides by -2, reversing the inequality sign:

x ≤ -1

Thus, x ∈ (-∞, -1]

Problem 4: Solve the inequality: -2 < 3x – 5 ≤ 7

Solution:

Given: -2 < 3x – 5 ≤ 7

Add 5 to all parts:

-2 + 5 < 3x – 5 + 5 ≤ 7 + 5

⇒ 3 < 3x ≤ 12

Divide all parts by 3:

1 < x ≤ 4

Thus, x ∈ (1, 4]

Problem 6: Solve the inequality: 5(2x – 1) ≤ 3(3x + 4)

Solution:

Given: 5(2x – 1) ≤ 3(3x + 4)

Distribute the constants:

10x – 5 ≤ 9x + 12

Subtract 9x from both sides:

10x – 9x – 5 ≤ 12

⇒ x – 5 ≤ 12

Add 5 to both sides:

x ≤ 17

Thus, x ∈ (-∞, 17]

Problem 7: Solve the inequality: 4x + 7 > 2x – 5

Solution:

Given: 4x + 7 > 2x – 5

Subtract 2x from both sides:

4x – 2x + 7 > 2x – 2x – 5

⇒ 2x + 7 > -5

Subtract 7 from both sides:

2x > -12

Divide both sides by 2:

x > -6

Thus, x ∈ (-6, ∞)

Problem 8: Solve the inequality: [Tex] \frac{2x – 3}{4} \leq 1[/Tex].

Solution:

Given: [Tex] \frac{2x – 3}{4} ≤ 1[/Tex]

Multiply both sides by 4 to eliminate the fraction:

[Tex]4 \cdot \frac{2x – 3}{4} ≤ 1 \cdot 4[/Tex]

⇒ 2x – 3 ≤ 4

Add 3 to both sides:

2x – 3 + 3 ≤ 4 + 3

⇒ 2x ≤ 7

Divide both sides by 2:

x ≤ 7/2

⇒ x ≤ 3.5

Thus, x ∈ (-∞, 3.5]

Problem 10: Solve the inequality: 3 < 2(x + 1) ≤ 7

Solution:

Given: 3 < 2(x + 1) ≤ 7

Divide the compound inequality into two parts and solve each separately.

Solve 3 < 2(x + 1):

Divide by 2:

3/2 < x + 1

Subtract 1 from both sides:

3/2 – 1 < x

⇒ 1/2 < x

Solve 2(x + 1) ≤ 7:

Divide by 2:

x + 1 ≤ 7/2

Subtract 1 from both sides:

x ≤ 5/2

Combine the solutions:

1/2 < x ≤ 5/2

Thus, x ∈ (1/2, 5/2]

Practice Problems on Linear Inequalities

Problem 1: Solve the inequality:

3x – 4 ≤ 2x + 5

Problem 2: Solve the compound inequality:

-2 ≤ 4 – 3x < 10

Problem 3: Solve the inequality and graph the solution:

(2x + 3)/5 > 1

Problem 4: Solve the system of inequalities:

• 2x + y ≥ 3
• x – y < 2

Problem 5: Solve the inequality:

5 – 2(x – 3) ≥ 3(x + 4) – 7

Problem 6: Solve the compound inequality:

-1 < 2x + 1 ≤ 5

Problem 7: Solve the inequality:

4(2x – 1) < 3(x + 6)

Problem 8: Solve the inequality:

7 – 3(2x + 1) ≥ 2(3 – x) + 1

Problem 9: Solve the inequality and represent the solution on a number line:

\frac{3x – 2}{4} ≤ \frac{x + 5}{2}

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: Linear Inequalities

What are linear inequalities?

Linear inequalities are mathematical expressions that show the relationship between two expressions using inequality symbols (<, ≤, >, ≥).

What is a compound inequality?

A compound inequality involves two or more inequalities joined by “and” or “or”. An “and” compound inequality means both conditions must be true simultaneously, while an “or” compound inequality means at least one of the conditions must be true.

How do you represent the solution of a linear inequality?

The solution of a linear inequality can be represented using interval notation, a number line, or a shaded region on a graph.

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

A linear equation uses an equal sign (=) to show that two expressions are equal, while a linear inequality uses inequality symbols (<, ≤, >, ≥) to show that one expression is less than, greater than, or equal to the other.

Can linear inequalities have more than one variable?

Yes, linear inequalities can have more than one variable. For example, 2x + 3y ≤ 6 is a linear inequality with two variables.

What does it mean to reverse the inequality sign?

When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign to maintain the correct relationship.

Can a system of linear inequalities have no solution?

Yes, a system of linear inequalities can have no solution if there is no region that satisfies all the inequalities simultaneously.