Solve inequality |2x-3| < 5 and represent solution on number line

Question

Solve the inequality $|2x - 3| < 5$ and represent the solution set on a number line. Write the solution in interval notation.

(NCERT Class 11, Linear Inequalities)

Solution — Step by Step

For any expression, $|A| < B$ (where $B > 0$ ) is equivalent to:

-B < A < B

Applying this to $|2x - 3| < 5$ :

-5 < 2x - 3 < 5

Add 3 to all three parts:

-5 + 3 < 2x < 5 + 3

-2 < 2x < 8

Divide all parts by 2:

-1 < x < 4

The solution set is $x \in \mathbf{(-1, 4)}$ .

This is an open interval — the endpoints $-1$ and $4$ are NOT included (because the inequality is strict: $<$ , not $\leq$ ).

On the number line:

Mark $-1$ and $4$ with open circles (hollow dots, since they’re not included)
Shade the region between $-1$ and $4$
The shaded region represents all real numbers strictly between $-1$ and $4$

Why This Works

The absolute value $|2x - 3|$ represents the distance of $2x - 3$ from 0 on the number line. Saying $|2x - 3| < 5$ means “the expression $2x - 3$ is less than 5 units away from 0.” That’s the same as saying $2x - 3$ lies between $-5$ and $+5$ .

This distance interpretation is powerful. Any absolute value inequality can be converted to a compound inequality using:

$|A| < B \iff -B < A < B$ (A is within B units of 0)
$|A| > B \iff A < -B$ or $A > B$ (A is more than B units away from 0)

Alternative Method

You can also split into two cases:

Case 1: If $2x - 3 \geq 0$ (i.e., $x \geq 1.5$ ), then $|2x - 3| = 2x - 3$ :

2x - 3 < 5 \implies x < 4

Combined with $x \geq 1.5$ : $1.5 \leq x < 4$

Case 2: If $2x - 3 < 0$ (i.e., $x < 1.5$ ), then $|2x - 3| = -(2x - 3) = 3 - 2x$ :

3 - 2x < 5 \implies -2x < 2 \implies x > -1

Combined with $x < 1.5$ : $-1 < x < 1.5$

Union: $(-1, 4)$ — same answer.

For “less than” modulus inequalities ( $|A| < B$ ), the solution is always a single interval (connected region). For “greater than” modulus inequalities ( $|A| > B$ ), the solution is always two disjoint intervals (two separate rays). This pattern helps you quickly verify your answer.

Common Mistake

Students sometimes write the answer as $x < 4$ and $x > -1$ separately, then combine them as $x < 4$ OR $x > -1$ (which would include all real numbers). The correct combination is $x < 4$ AND $x > -1$ , written as $-1 < x < 4$ . With “less than” modulus inequalities, the two conditions must BOTH be satisfied — it’s an intersection, not a union.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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