Compound and Absolute Value Inequalities

Tags: compound inequalities, absolute value, number lines, inequalities, 7.EE.B.4, A.REI.B.3

Concept Overview

This page explores two important types of inequalities: compound inequalities and absolute value inequalities. Both involve comparisons that lead to ranges of values rather than single-number solutions. Interpreting these correctly is essential for solving real-world and algebraic problems.

Key Vocabulary

Compound Inequality: Two inequalities joined by “and” or “or”
Absolute Value: The distance from zero on the number line, always nonnegative
Boundary: A value that separates regions of solutions from non-solutions
Inclusive: Includes the boundary (\( \leq, \geq \))
Exclusive: Does not include the boundary (\( <, > \))

Understanding Compound Inequalities

A compound inequality includes two inequalities joined by the word and or or. These statements describe sets of values that meet either both conditions at once (intersection) or at least one of the conditions (union).

“And” Compound Inequalities

These represent values that satisfy both parts of the inequality. The solution is the overlap of the two conditions.

\( -3 < x \leq 2 \)

This represents all \( x \) values between -3 and 2, including 2 but not -3.

“Or” Compound Inequalities

These represent values that satisfy either one condition or the other (or both). The solution is the union of the regions.

\( x < -2 \quad \text{or} \quad x \geq 3 \)

This means \( x \) can be any value less than -2 or any value 3 or greater. There is a break between the two regions.

Note: Always pay attention to whether endpoints are included (closed circle) or excluded (open circle).

Solving Absolute Value Inequalities

An absolute value inequality compares the distance between a number and zero. Because absolute value is always nonnegative, these inequalities often lead to two-part compound inequalities.

Case 1: “Less than” form

\( |x – 2| < 5 \)

This means the distance between \( x \) and 2 is less than 5. It can be rewritten as a conjunction:

\( -5 < x – 2 < 5 \)

Add 2 to all parts:

\( -3 < x < 7 \)

So, the solution is all values between -3 and 7 (excluding endpoints).

Graph of \( -3 < x < 7 \): open interval with endpoints not included.

Case 2: “Greater than” form

\( |x + 1| \geq 4 \)

This means the distance between \( x \) and -1 is at least 4. It creates a disjunction (two separate cases):

\( x + 1 \leq -4 \quad \text{or} \quad x + 1 \geq 4 \)

Subtract 1 from both sides:

\( x \leq -5 \quad \text{or} \quad x \geq 3 \)

The solution includes all values less than or equal to -5, or greater than or equal to 3.

Graph of \( x \leq -5 \) or \( x \geq 3 \): shaded beyond both bounds with closed circles.

Tip: Remember:

“Less than” (\( < \)) → “and” → overlap
“Greater than” (\( > \)) → “or” → two separate regions

Try It Yourself!

Solve and graph: \( -6 \leq x < 1 \)
Solve and graph: \( x < -2 \ \text{or} \ x \geq 5 \)
Solve and graph: \( |x – 4| < 3 \)
Solve and graph: \( |x + 2| \geq 6 \)

Reveal Answers

\( -6 \leq x < 1 \)
\( x < -2 \quad \text{or} \quad x \geq 5 \)
\( -3 < x – 4 < 3 \)
Add 4 to all parts:
\( 1 < x < 7 \)
\( |x + 2| \geq 6 \Rightarrow x \leq -8 \ \text{or} \ x \geq 4 \)

Connections & Extensions

Inequalities model constraints and tolerance in real-world problems, such as acceptable ranges in engineering, data thresholds, or budgeting. Understanding compound and absolute value forms is essential for interpreting boundaries in more advanced contexts like systems, piecewise functions, and optimization.