Back to Lesson

Teacher Guide: Absolute Value Inequalities

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

Present LessonView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Inequalities. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Interpret absolute value as distance from zero on a number line
  • Solve absolute value inequalities of the form and
  • Write solutions in interval notation
  • Identify special cases with no solution or all real numbers
  • Apply absolute value inequalities to real-world tolerance problems
Prerequisites
  • • Understanding of absolute value as distance from zero
  • • Solving linear inequalities
  • • Compound inequalities (AND and OR)
  • • Interval notation
Discussion Starters
  • 1. Why does give values BETWEEN and , while gives values OUTSIDE that range?
  • 2. Can you think of a job where understanding tolerances and error margins is important?
  • 3. If , how close is to 5? What real-world situation might require such precision?
  • 4. Why is it impossible for to have any solutions?
Common Misconceptions

Treating the same as

Thinking absolute value can be negative

Forgetting to solve BOTH parts of a greater-than inequality

Differentiation Ideas

For Struggling Students:

  • • Start with simple cases like before adding expressions
  • • Use colored number lines to show the two regions
  • • Provide a reference card with the two formulas

For On-Level Students:

  • • Solve inequalities with linear expressions inside absolute value
  • • Write solutions in both inequality and interval notation
  • • Apply to word problems involving tolerances

For Advanced Students:

  • • Explore absolute value inequalities with variables on both sides
  • • Investigate graphical solutions using
  • • Solve more complex inequalities like
Standards Alignment
  • HSA-REI.B.3 (CCSS.MATH.CONTENT.HSA.REI.B.3)

    Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters

  • HSA-CED.A.1 (CCSS.MATH.CONTENT.HSA.CED.A.1)

    Create equations and inequalities in one variable and use them to solve problems

Lesson Resources
  • visualInteractive Number Line

    Visualize solution sets for absolute value inequalities

  • activityTolerance Challenge

    Match real-world scenarios to their absolute value inequalities

  • worksheetPractice Problems

    Mixed practice with all types of absolute value inequalities

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

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

Two Main Types

Less Than ( or ): The expression is within a certain distance from zero.
Greater Than ( or ): The expression is beyond a certain distance from zero.

Key Insight

  • means is less than 3 units from zero:
  • means is more than 3 units from zero: or

Worked Examples

Solve

1

Recognize the type

is a "less than" inequality → Expression is within 5 units of zero

2

Write as compound inequality

For , write →

3

Solve by adding 2 to all parts

→

4

Write solution set

All numbers between and →

Common Mistakes

Writing as or

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

Correct: means (values between and )

Writing as

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

Correct: means OR (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: Be careful when the coefficient of is negative inside the absolute value

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

Why it's wrong: Absolute value is always non-negative, so has no solution, and is always true.

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

Why It Matters

Absolute value inequalities are essential in many real-world applications:
  • Quality Control: A machine part must be within mm of the target size:
  • Temperature Ranges: Keep medicine within 3 degrees of 20 degrees Celsius:
  • 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 means the diameter 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: Write the mathematical expression

If is the length, the tolerance inequality is:

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: , so 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: Write the mathematical expression

Write as:

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: .

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: Write the mathematical expression

Check if

Key Takeaways

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

Frequently Asked Questions

How do I remember when to use AND vs OR?

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 = beLEss = betweenLESS = between.

What if the absolute value equals a number, not less than or greater than?

For , you get exactly two solutions: or . This is an equation, not an inequality.

Can the number on the right side be negative?

Yes, but check carefully! Since absolute value is always non-negative: has no solution, while is true for all values.

Glossary

Absolute value
The distance of a number from zero on the number line, always non-negative. Written as
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

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

Greater Than (or Equal) Case

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

More in This Topic

Introduction to Inequalities

Intermediate20 min

Solving and Graphing Inequalities

Intermediate25 min

Graphing Inequalities on a Number Line

Intermediate25 min

One-Step Inequalities

Intermediate25 min