Class 9 Mathematics Exercise 5.3 – Absolute Value Equations Solved Step-by-Step

Exercise 5.3 from Class 9 Mathematics introduces students to one of the most important concepts in algebra: absolute value equations. Many students initially find these questions confusing because absolute value expressions create two possible cases. However, once you understand the basic rule, solving these equations becomes straightforward and logical.

In this lesson, we explain every important concept related to absolute value equations with clear definitions, solved examples, algebraic techniques, and exam-focused explanations. This topic is extremely important for FBISE, Matric, and O-Level students because it builds a strong foundation for higher algebra, functions, inequalities, and coordinate geometry.

Class 9 Maths Exercise 5.3 Linear Equations and Inequalities showing algebraic expressions, inequality signs, and step-by-step solutions for solving linear inequality problems

What Is Absolute Value?

The absolute value of a number represents its distance from zero on the number line. Since distance can never be negative, the absolute value of any number is always positive or zero.

Examples:

|5| = 5
|-5| = 5
|0| = 0

Both positive and negative values can have the same absolute value because they are equally distant from zero. This is the key idea used while solving absolute value equations.

Main Rule for Solving Absolute Value Equations

Whenever you solve an equation of the form:

|x| = a

you must consider two possibilities:

x = a
x = -a

This happens because both positive and negative numbers can have the same absolute value.

Example:

|x| = 7

Therefore:

x = 7
x = -7

Final Answer: x = ±7

The Special Zero Case

One important rule that students often forget in exams is the zero case.

If:

|x| = 0

then:

x = 0 only

We do not write ±0 because positive and negative zero are the same value.

Solved Example 1

Solve: |x + 2| = 6

According to the absolute value rule:

x + 2 = 6 or x + 2 = -6

Case 1:

x + 2 = 6
x = 6 - 2
x = 4

Case 2:

x + 2 = -6
x = -6 - 2
x = -8

Final Answer: x = 4 or x = -8

Solved Example 2

Solve: |5y - 1| = 9

We create two equations:

5y - 1 = 9
5y - 1 = -9

Case 1:

5y - 1 = 9
5y = 10
y = 2

Case 2:

5y - 1 = -9
5y = -8
y = -8/5

Final Answer: y = 2 or y = -8/5

Solved Example 3 – Equation with Coefficients

Solve: 3|z - 2| - 4 = -2

First isolate the absolute value term:

3|z - 2| = 2
|z - 2| = 2/3

Now create two cases:

z - 2 = 2/3
z - 2 = -2/3

Case 1:

z = 2 + 2/3
z = 8/3

Case 2:

z = 2 - 2/3
z = 4/3

Final Answer: z = 8/3 or z = 4/3

Absolute Value Equations with Fractions

Students often struggle with fractional equations, but the method remains the same.

Example:

|4x/3| = 12

Multiply both sides by 3:

|4x| = 36

Now solve:

4x = 36
4x = -36

x = 9
x = -9

Final Answer: x = ±9

Equations with Absolute Values on Both Sides

Some advanced questions contain variables inside absolute values on both sides of the equation. These questions require careful algebraic manipulation.

Example:

|x + 1| / 2 = |2x - 1| / 3

Cross multiply:

3|x + 1| = 2|2x - 1|

Now solve using:

3(x + 1) = ±2(2x - 1)

After solving both cases, we get:

x = 5
x = -1/7

These questions are important for board exams because they test both algebraic understanding and logical thinking.

Common Mistakes Students Make

Forgetting the negative case while solving absolute value equations
Making sign errors during simplification
Not isolating the absolute value before solving
Incorrectly applying the ± sign to zero
Skipping algebraic steps and losing marks in board exams

During exam preparation, always solve each case separately and write every step clearly. This improves accuracy and helps secure full marks.

Why Absolute Value Equations Matter

Absolute value equations are not just textbook exercises. They are widely used in higher mathematics, computer science, engineering, physics, and coordinate geometry.

This chapter also improves logical reasoning because students learn how one equation can produce multiple valid solutions. Strong understanding of this topic helps students perform better in algebraic problem-solving throughout higher classes.

Exam Tips for Exercise 5.3

Always isolate the absolute value term first
Remember that |x| = a gives two cases
Write positive and negative cases separately
Check your answers by substitution
Practice fraction-based and multi-step equations regularly

Most students lose marks because they skip steps or forget one of the solutions. Careful working and proper algebraic simplification are essential for high marks in FBISE and Matric examinations.