Absolute Value Inequalities Calculator

Solve absolute value inequalities. Get solutions in standard form, interval notation, and graphical descriptions.

The value being subtracted from x in |x - b|

The bound value (must be positive)

**Less Than Inequalities:** |x - b| < a (where a > 0) Solution: -a < x - b < a Simplified: b - a < x < b + a Interval: (b-a, b+a) |x - b| = a (where a > 0) Solution: -a = x - b = a Simplified: b - a = x = b + a Interval: [b-a, b+a] **Greater Than Inequalities:** |x - b| > a (where a > 0) Solution: x - b < -a OR x - b > a Simplified: x < b - a OR x > b + a Interval: (-8, b-a) ? (b+a, 8) |x - b| = a (where a > 0) Solution: x - b = -a OR x - b = a Simplified: x = b - a OR x = b + a Interval: (-8, b-a] ? [b+a, 8) **Key Concept:** Absolute value |x - b| represents the distance from x to b. � < or = creates a single interval (between two points) � > or = creates two separate intervals (outside two points)
**Example 1: Less Than** Solve: |x - 3| < 5 Step 1: Identify b = 3, a = 5 Step 2: Apply formula: -5 < x - 3 < 5 Step 3: Add 3 to all parts: -5 + 3 < x < 5 + 3 Step 4: Simplify: -2 < x < 8 Solution: -2 < x < 8 Interval Notation: (-2, 8) Interpretation: x is within 5 units of 3 Graph: Open circles at -2 and 8, shaded between **Example 2: Greater Than or Equal** Solve: |x + 2| = 4 Step 1: Rewrite: |x - (-2)| = 4, so b = -2, a = 4 Step 2: Split into two cases: x - (-2) = -4 OR x - (-2) = 4 Step 3: Simplify: x + 2 = -4 OR x + 2 = 4 Step 4: Solve each: x = -6 OR x = 2 Solution: x = -6 OR x = 2 Interval Notation: (-8, -6] ? [2, 8) Interpretation: x is at least 4 units away from -2 Graph: Closed circles at -6 and 2, shaded left of -6 and right of 2

What is an absolute value inequality?

An inequality containing absolute value expressions like |x-3| < 5 or |2x+1| >= 7. Absolute value |a| measures distance from zero. |x-3| < 5 means "distance from 3 is less than 5", so -2 < x < 8. Used to express ranges and error tolerances.

How do you solve |x| < a inequalities?

For |x| < a (where a > 0): -a < x < a (between -a and a). Example: |x| < 5 means -5 < x < 5. For |x-b| < a: -a < x-b < a, so b-a < x < b+a. This creates an "and" compound inequality (interval notation: (b-a, b+a)).

How do you solve |x| > a inequalities?

For |x| > a (where a > 0): x < -a OR x > a (outside the interval). Example: |x| > 5 means x < -5 or x > 5. For |x-b| > a: x-b < -a OR x-b > a, so x < b-a or x > b+a. This creates an "or" compound inequality (two separate rays).

What is the difference between < and <= in absolute value inequalities?

< (strict): endpoints NOT included. Example: |x| < 5 → -5 < x < 5 → (-5, 5). <= (non-strict): endpoints included. Example: |x| <= 5 → -5 <= x <= 5 → [-5, 5]. On number line: open circles for <, closed circles for <=. Same for > vs >=.

What are real-world applications of absolute value inequalities?

Manufacturing tolerances: |actual - target| <= tolerance. Temperature ranges: |temp - 72| <= 5 means 67degF to 77degF. Error margins: |measured - true| < 0.01. Quality control: acceptable deviation from standard. Physics: uncertainty in measurements. Any scenario requiring distance or deviation bounds.