Absolute Value Inequalities

Learn to solve inequalities involving absolute value by understanding distance from zero.

Advanced25 min17 problems
Practice Problems

Definition

An absolute value inequality involves the absolute value of an expression compared to a number.
Recall that ∣x∣|x| represents the distance from xx to zero on the number line.

Two Main Types

Less Than (<< or ≤\leq): The expression is within a certain distance from zero.
∣x∣<a⇒−a<x<a|x| < a \Rightarrow -a < x < a
Greater Than (>> or ≥\geq): The expression is beyond a certain distance from zero.
∣x∣>a⇒x<−a or x>a|x| > a \Rightarrow x < -a \text{ or } x > a

Key Insight

  • ∣x∣<3|x| < 3 means xx is less than 3 units from zero: −3<x<3-3 < x < 3
  • ∣x∣>3|x| > 3 means xx is more than 3 units from zero: x<−3x < -3 or x>3x > 3

Try it now

Which compound inequality represents ∣x∣<5|x| < 5?

Worked Examples

Solve ∣x−2∣<5|x - 2| < 5
1

Recognize the type

∣x−2∣<5|x - 2| < 5 is a "less than" inequality → Expression is within 5 units of zero
2

Write as compound inequality

For ∣A∣<a|A| < a, write −a<A<a-a < A < a → −5<x−2<5-5 < x - 2 < 5
3

Solve by adding 2 to all parts

−5+2<x−2+2<5+2-5 + 2 < x - 2 + 2 < 5 + 2 → −3<x<7-3 < x < 7
4

Write solution set

All numbers between −3-3 and 77 → x∈(−3,7)x \in (-3, 7)

Common Mistakes

Writing ∣x∣<3|x| < 3 as x<3x < 3 or x<−3x < -3

Why it's wrong: Students forget that "less than" creates an AND compound inequality (between two values), not an OR.

Correct: ∣x∣<3|x| < 3 means −3<x<3-3 < x < 3 (values between −3-3 and 33)

Writing ∣x∣>3|x| > 3 as −3<x<3-3 < x < 3

Why it's wrong: Confusing the two cases. "Greater than" means the value is FAR from zero, not close to it.

Correct: ∣x∣>3|x| > 3 means x<−3x < -3 OR x>3x > 3 (values outside the interval)

Forgetting to flip the inequality when multiplying by a negative

Why it's wrong: While solving, if you multiply or divide by a negative number, the inequality direction reverses.

Correct: Example: ∣3−2x∣<5|3 - 2x| < 5 gives −5<3−2x<5-5 < 3 - 2x < 5, so −8<−2x<2-8 < -2x < 2. Dividing by −2-2 flips both signs: 4>x>−14 > x > -1, that is −1<x<4-1 < x < 4

Not checking if the inequality has no solution or all real numbers

Why it's wrong: Absolute value is always non-negative, so ∣A∣<negative|A| < \text{negative} has no solution, and ∣A∣>negative|A| > \text{negative} is always true.

Correct: Always check: Can this inequality actually be satisfied?

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Solve Absolute Value Inequalities | Step-by-Step “Less Than” and “Greater Than” Cases

by Understand The Math

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Interactive Visual

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-1 ≤ x ≤ 5·[-1, 5]

Click on numbers to select them. Adjust the range to explore different values.

to

x < -4 or x > 3·(−∞, -4) ∪ (3, ∞)

Click on numbers to select them. Adjust the range to explore different values.

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Practice Problems

17 problems
Problem 1 of 17
Easy
Which compound inequality represents ∣x∣<5|x| < 5?

Why It Matters

Absolute value inequalities are essential in many real-world applications:
  • Quality Control: A machine part must be within ±0.02\pm 0.02 mm of the target size: ∣x−50∣≤0.02|x - 50| \leq 0.02
  • Temperature Ranges: Keep medicine within 3 degrees of 20 degrees Celsius: ∣T−20∣≤3|T - 20| \leq 3
  • Statistics: Values within 2 standard deviations of the mean
  • Engineering: Tolerance ranges for measurements and specifications
Understanding these inequalities helps you express acceptable ranges and error margins mathematically.

Real World Applications

Manufacturing Tolerances

Factory machines must produce parts within specific tolerances. A bolt must be 10 mm with tolerance of 0.05 mm.

Example:

The inequality ∣d−10∣≤0.05|d - 10| \leq 0.05 means the diameter dd must be between 9.95 mm and 10.05 mm.

1Try It Yourself

A factory produces rods that must be 25 cm long, with an acceptable error of 0.2 cm.

Write and solve the inequality for acceptable rod lengths.

Step 1: Choose the right answer

Which inequality describes the acceptable lengths LL?

Temperature Control

Many products require storage within specific temperature ranges.

Example:

Vaccines must be stored at 4 degrees Celsius with variation no more than 2 degrees: ∣T−4∣≤2|T - 4| \leq 2, so 2≤T≤62 \leq T \leq 6 degrees.

2Try It Yourself

A wine cellar maintains temperature at 13 degrees Celsius. The temperature must not vary by more than 1.5 degrees.

What is the acceptable temperature range?

Step 1: Choose the right answer

Which inequality describes the allowed temperatures TT?

Speed Limits and Radar

Police radar has a margin of error. A driver might contest a ticket if the measured speed is within the error margin.

Example:

If radar accuracy is 3 km/h and the limit is 50 km/h, speeds registering between 47 and 53 might be within error: ∣v−50∣≤3|v - 50| \leq 3.

3Try It Yourself

A speed camera has an error margin of 2 km/h. A driver was recorded at 62 km/h in a 60 km/h zone.

Is the recorded speed within the error margin of the limit?

Step 1: Choose the right answer

Which statement is true for this driver?

Key Takeaways

  • 1Absolute value measures distance from zero, so ∣x∣|x| is always non-negative
  • 2For ∣A∣<a|A| < a (less than): write −a<A<a-a < A < a (AND compound inequality)
  • 3For ∣A∣>a|A| > a (greater than): write A<−aA < -a OR A>aA > a (OR compound inequality)
  • 4If comparing to a negative number: ∣A∣<negative|A| < \text{negative} has no solution; ∣A∣>negative|A| > \text{negative} is all real numbers
  • 5Always isolate the absolute value expression before splitting into cases

Frequently Asked Questions

Think of distance: "less than" means CLOSE to zero (between two values = AND). "Greater than" means FAR from zero (outside = OR). Memory trick: less than gives one piece in the middle, greater than gives two pieces on the outside.
Think of distance: "less than" means CLOSE to zero (between two values = AND). "Greater than" means FAR from zero (outside = OR). Memory trick: less than gives one piece in the middle, greater than gives two pieces on the outside.
For ∣A∣=a|A| = a, you get exactly two solutions: A=aA = a or A=−aA = -a. This is an equation, not an inequality.
Yes, but check carefully! Since absolute value is always non-negative: ∣A∣<negative|A| < \text{negative} has no solution, while ∣A∣>negative|A| > \text{negative} is true for all values.

Glossary

Absolute value
The distance of a number from zero on the number line, always non-negative. Written as ∣x∣|x|
Compound inequality
Two inequalities joined by AND or OR
Interval notation
A way to write solution sets using parentheses (exclude endpoint) and brackets (include endpoint)
Solution set
The set of all values that make an inequality true

Formula Card

Less Than (or Equal) Case

∣A∣<a⇒−a<A<a|A| < a \Rightarrow -a < A < a

The expression $A$ is within $a$ units of zero

Greater Than (or Equal) Case

∣A∣>a⇒A<−a or A>a|A| > a \Rightarrow A < -a \text{ or } A > a

The expression $A$ is more than $a$ units from zero

Today's challengeOne problem, three tries

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