5-4 Solving Absolute Value Inequalities

Property

The expression $|x - c|$ algebraically represents the exact distance between a number $x$ and a center point $c$ on the number line.
Because it represents distance, absolute value is always non-negative. Using this geometric interpretation:

• $|x - c| < a$ means the distance from $x$ to $c$ is strictly less than $a$.
• $|x - c| > a$ means the distance from $x$ to $c$ is strictly greater than $a$.

Examples

• The inequality $|x - 3| < 5$ means the distance from $x$ to 3 is less than 5. Therefore, $x$ must lie within 5 units of 3 on the number line, which is anywhere between -2 and 8.
• The inequality $|x + 2| > 4$ can be rewritten as $|x - (-2)| > 4$. This means the distance from $x$ to -2 is greater than 4. So, $x$ must be more than 4 units away from -2, putting it to the left of -6 or to the right of 2.
• The inequality $|x| \leq 7$ means the distance from $x$ to 0 is at most 7, giving the solution $-7 \leq x \leq 7$.

Explanation

Absolute value is not just a rule that "makes numbers positive"; it is a mathematical measuring tape. It measures how far a number is from a specific center point. This geometric view turns a confusing algebraic inequality into a simple question of distance: "Which numbers are within a certain range of my target, and which numbers are too far away?" Building this intuition makes the algebraic steps that follow feel natural rather than mechanical.

Property

Before you can split an absolute value inequality into different cases, you must isolate the absolute value expression on one side of the inequality symbol.

Once isolated, evaluate the number on the other side. Because absolute value represents distance, it cannot be negative:

• If $|expression| < \text{negative number}$, there is No Solution ($\emptyset$).
• If $|expression| > \text{negative number}$, the solution is All Real Numbers $(-\infty, \infty)$.

Examples

• Isolate First: Solve $2|x - 3| - 4 < 6$.

Add 4 to both sides: $2|x - 3| < 10$.
Divide by 2: $|x - 3| < 5$. (Now it is ready to be solved using cases).

• Special Case (No Solution): Solve $|x + 4| + 5 < 2$.

Subtract 5 from both sides: $|x + 4| < -3$.
Since a distance cannot be less than a negative number, there is No Solution.

• Special Case (All Real Numbers): Solve $|x - 9| > -2$.

Since an absolute value is always 0 or positive, it will always be greater than -2. Therefore, the solution is All Real Numbers.

Property

When an isolated absolute value inequality uses a "less than" ($<$) or "less than or equal to" ($\leq$) symbol with a positive number, it translates into a compound "AND" inequality.
If $|u| < a$, the equivalent compound inequality is $-a < u < a$.
This means the expression inside must be trapped between the negative and positive boundaries of that distance.

Examples

• Simple Less-Than: Solve $|x| < 9$.

You are looking for all numbers whose distance from zero is less than 9. Rewrite as a compound inequality: $-9 < x < 9$. The solution is $(-9, 9)$.

• Multi-Step Less-Than: Solve $|2x - 5| \leq 3$.

Rewrite as a compound "and" inequality: $-3 \leq 2x - 5 \leq 3$.
Add 5 to all three parts: $2 \leq 2x \leq 8$.
Divide all three parts by 2: $1 \leq x \leq 4$. The solution is $[1, 4]$.

Explanation

When an absolute value is "less than" a number, it means the expression is constrained. It is kept close to the center point. To solve it algebraically, you drop the absolute value bars and sandwich the inner expression between the negative and positive limits. You then solve all three parts of the chain at the same time to find the exact overlapping region.