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Absolute Value Inequalities

Inequalities like |x - a| < b or |x - a| > b split into cases: for <, the solution is -b < expr < b; for >, it is expr < -b or expr > b.

Concept Fundamentals
|x| < c → -c < x < c
Less than
|x| > c → x < -c or x > c
Greater than
Distance from origin
Geometric
Two cases to solve
Split

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|x - a| < b means x is within b units of a. For |expr| > 0, solution is all reals except where expr = 0. Number line visualization shows the solution intervals.

Key quantities
|x| < c → -c < x < c
Less than
Key relation
|x| > c → x < -c or x > c
Greater than
Key relation
Distance from origin
Geometric
Key relation
Two cases to solve
Split
Key relation

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Why: Absolute value inequalities model distance constraints and tolerance ranges in engineering and optimization.

How: Split into two inequalities: remove the absolute value and solve each branch, then combine with AND or OR.

|x - a| < b means x is within b units of a.For |expr| > 0, solution is all reals except where expr = 0.
Sources:Khan AcademyOpenStax Algebra

Run the calculator when you are ready.

Solve Absolute Value InequalitiesEnter |expression| < or > value

Absolute Value Inequalities Solver

Format: |expression| operator value (e.g., |x - 3| < 5, |2x + 1| >= 7)

Step-by-Step Solution

-5 < x - 3 < 5

-5 + 3 < x < 5 + 3

-2 < x < 8

Solution

-2 < x < 8

Interval Notation

(-2, 8)

Number Line Visualization

0
-5
-4
-3
-2
-1
1
2
3
4
5

Solution visualized on the number line (approximate scale)

Understanding Absolute Value Inequalities

An absolute value inequality is an inequality that contains an absolute value expression. These inequalities typically have solutions that are intervals or unions of intervals on the number line.

The Basic Forms

Less Than Form

|expression| < c

|expression| ≤ c

Greater Than Form

|expression| > c

|expression| ≥ c

Solving Absolute Value Inequalities

To solve absolute value inequalities, we need to consider different cases based on the type of inequality and the value of c:

When |expression| < c or |expression| ≤ c (c > 0)

This means the expression's distance from zero is less than c, which is equivalent to:

-c < expression < c (or with ≤ if original has ≤)

The solution is a bounded interval, typically written as (-c, c) or [-c, c] depending on whether the original inequality is strict (<) or not (≤).

When |expression| > c or |expression| ≥ c (c > 0)

This means the expression's distance from zero is greater than c, which is equivalent to:

expression < -c or expression > c (or with ≥ if original has ≥)

The solution is a union of two unbounded intervals, typically written as (-∞, -c) ∪ (c, ∞) or (-∞, -c] ∪ [c, ∞) depending on the original inequality.

Special Cases for c ≤ 0

  • When c = 0:
    • If |expression| < 0, there is no solution (∅) since absolute values are never negative.
    • If |expression| ≤ 0, the solution is expression = 0.
    • If |expression| > 0, the solution is all real numbers except 0.
    • If |expression| ≥ 0, the solution is all real numbers (since absolute values are always ≥ 0).
  • When c < 0:
    • If |expression| < c or |expression| ≤ c, there is no solution (∅) since absolute values are never negative.
    • If |expression| > c or |expression| ≥ c, the solution is all real numbers (ℝ or (-∞, ∞)).

Example Problems

Example 1: Less Than Inequality

Solve |x - 3| < 5

Step 1: Rewrite the inequality:

-5 < x - 3 < 5

Step 2: Solve for x:

-5 + 3 < x < 5 + 3

-2 < x < 8

Solution: x ∈ (-2, 8)

Example 2: Greater Than Inequality

Solve |2x + 1| ≥ 7

Step 1: Rewrite as a union:

2x + 1 ≥ 7 or 2x + 1 ≤ -7

Step 2: Solve both inequalities:

2x ≥ 6 or 2x ≤ -8

x ≥ 3 or x ≤ -4

Solution: x ∈ (-∞, -4] ∪ [3, ∞)

Example 3: With a coefficient

Solve |3x - 6| ≤ 9

Step 1: Rewrite the inequality:

-9 ≤ 3x - 6 ≤ 9

Step 2: Solve for x:

-9 + 6 ≤ 3x ≤ 9 + 6

-3 ≤ 3x ≤ 15

-1 ≤ x ≤ 5

Solution: x ∈ [-1, 5]

Example 4: Special Case

Solve |x + 4| > -2

Step 1: Since absolute value is always ≥ 0, and -2 < 0, the absolute value is always greater than -2.

Solution: All real numbers, x ∈ (-∞, ∞)

Applications of Absolute Value Inequalities

Error Bounds and Tolerances

In engineering and manufacturing, absolute value inequalities describe acceptable deviations from target values. For example, |x - 10| ≤ 0.05 means the measurement x can be between 9.95 and 10.05.

Distance Problems

The inequality |x - a| < d represents all points whose distance from a is less than d. In higher dimensions, this defines regions like circles and spheres.

Signal Processing

Absolute value inequalities can describe signal thresholds and noise limits in signal processing and control systems.

Economics

Absolute value inequalities can model acceptable price fluctuations, risk margins, and other economic indicators where deviation from a target is constrained.

Tips for Solving Absolute Value Inequalities

  • Isolate the absolute value expression first, if necessary. For example, in 2|x - 3| - 1 < 5, first isolate to |x - 3| < 3.
  • Pay attention to the direction of the inequality. Less than (<) inequalities give bounded intervals, while greater than (>) inequalities give unbounded intervals.
  • Watch out for special cases when the constant is zero or negative.
  • Check your solutions by substituting test values from each interval to verify they satisfy the original inequality.
  • Remember interval notation: Use parentheses ( ) for strict inequalities and square brackets [ ] for non-strict inequalities.

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

|x| < c ⇔ -c < x < c for c > 0

⚡

|x| > c ⇔ x < -c or x > c

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