Absolute Value Inequalities Calculator
Solve absolute value inequalities with steps
How to Use
- Select the inequality type (< or >)
- Enter the value 'a' in |x - a|
- Enter the constant 'b' (must be positive)
- Click calculate to see the solution and interval notation
What are Absolute Value Inequalities?
Absolute value inequalities involve the absolute value of a variable expression compared to a constant. The absolute value |x| represents the distance from zero on a number line, always positive or zero.
Two Main Types
Less Than: |x - a| < b means x is within b units of a
Greater Than: |x - a| > b means x is more than b units away from a
How to Solve Absolute Value Inequalities
Type 1: |x - a| < b
This creates a compound inequality:
Step 1: Write as -b < x - a < b
Step 2: Add a to all parts: -b + a < x < b + a
Example: |x - 3| < 5 becomes -5 < x - 3 < 5, so -2 < x < 8
Type 2: |x - a| > b
This creates two separate inequalities:
Step 1: Write as x - a < -b OR x - a > b
Step 2: Solve each: x < a - b OR x > a + b
Example: |x - 3| > 5 gives x < -2 OR x > 8
Interval Notation
Interval notation provides a compact way to express solution sets:
Symbols
- (a, b) - Open interval: values between a and b, not including endpoints
- [a, b] - Closed interval: values between a and b, including endpoints
- ∪ - Union: combines two or more intervals
- ∞ - Infinity: extends without bound
Examples
-2 < x < 8 → (−2, 8)
x < -2 or x > 8 → (−∞, −2) ∪ (8, ∞)
Real-World Applications
Absolute value inequalities model situations involving tolerance, error margins, and acceptable ranges:
Common Uses
- Manufacturing: Part dimensions must be within tolerance (e.g., |d - 5| < 0.02 cm)
- Temperature Control: Room temperature maintained within a range
- Quality Control: Product weights within acceptable limits
- Physics: Measurement uncertainty and error bounds
- Statistics: Confidence intervals and standard deviations
Frequently Asked Questions
- What's the difference between |x - a| < b and |x - a| > b?
- Less than (<) gives a single continuous interval of values between two bounds, while greater than (>) gives two separate intervals extending outward from two bounds. Think of < as 'close to a' and > as 'far from a'.
- Why must the constant b be positive?
- Since absolute values are always non-negative, comparing them to a negative number creates impossible (<) or always-true (>) statements. A positive constant ensures meaningful solutions.
- How do I know which type of inequality to use?
- Use < when you want values within a certain distance from a point, and use > when you want values outside a certain distance. For example, temperature within ±5° uses <, while avoiding a danger zone uses >.
- What does the interval notation (a, b) mean?
- Parentheses indicate open intervals that don't include the endpoints. So (−2, 8) means all numbers between −2 and 8, but not −2 or 8 themselves. Brackets [ ] would include the endpoints.