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Compound
Inequalities

Learn to solve and graph compound inequalities that combine two conditions with "and" or "or", and interpret what the solutions mean.

Pre-Algebra

What Is an Inequality?

An inequality is a mathematical statement that compares two expressions using symbols like \( < \), \( > \), \( \leq \), or \( \geq \). Unlike an equation, an inequality can have many solutions instead of just one.

\( x > 3 \)
All numbers greater than 3
\( x \leq 7 \)
All numbers less than or equal to 7

Two Conditions, One Statement

A compound inequality joins two simple inequalities with the word "and" or "or". Think of it like a filter: you are describing which numbers pass both tests, or at least one test.

"And" Compound
\( x > 2 \) and \( x < 8 \)
Written: \( 2 < x < 8 \)
"Or" Compound
\( x < -1 \) or \( x > 5 \)
Two separate regions

Graphing on a Number Line

We show inequality solutions on a number line. Open circles mean the endpoint is not included; closed (filled) circles mean it is included.

"And": \( 2 < x < 8 \) 0 2 4 6 8 "Or": \( x < -1 \) or \( x > 5 \) -1 2 5

"And" Means Intersection

An "and" compound inequality requires a number to satisfy both conditions at the same time. The solution set is the overlap (intersection) of the two individual solutions.

\( x > 1 \) and \( x < 6 \) \( x > 1 \) 1 \( x < 6 \) 6 Overlap

"Or" Means Union

An "or" compound inequality requires a number to satisfy at least one condition. The solution set is the union of the two individual solutions: everything from either part.

\( x < -2 \) or \( x > 4 \) \( x < -2 \) -2 \( x > 4 \) 4 Union

And vs Or: The Key Difference

"And" (Intersection) "Or" (Union)
Meaning Both conditions true At least one condition true
Graph Shape Segment between endpoints Two rays going outward
Compact Form \( a < x < b \) \( x < a \) or \( x > b \)
Example \( -3 \leq x \leq 5 \) \( x < -3 \) or \( x > 5 \)
Narrows or Widens? Narrows solutions Widens solutions

Open vs Closed Circles

The type of circle at an endpoint tells you whether that boundary value is included in the solution set.

Open Circle
Used with \( < \) or \( > \). The endpoint is not a solution.
Closed Circle
Used with \( \leq \) or \( \geq \). The endpoint is a solution.
\( x > 3 \) (open) 3 \( x \geq 3 \) (closed)

How to Solve Compound Inequalities

1
Identify the type: Does the problem use "and" or "or"?
2
Solve each inequality separately. Use inverse operations just like a regular equation.
3
Remember the flip rule: If you multiply or divide by a negative number, reverse the inequality sign.
4
Combine the solutions: "And" = overlap. "Or" = union.
5
Graph it: Draw open/closed circles and shade the correct region on a number line.

Example 1: Solving an "And" Inequality

1
Subtract 2 from all three parts: \( -4 - 2 < 2x < 10 - 2 \) gives \( -6 < 2x < 8 \)
2
Divide all parts by 2: \( -3 < x < 4 \)
-3 4

Example 2: Solving an "Or" Inequality

Left part
\( 3x - 1 < -7 \)
\( 3x < -6 \)
\( x < -2 \)
Right part
\( 3x - 1 > 5 \)
\( 3x > 6 \)
\( x > 2 \)
-2 2

Example 3: The Flip Rule

1
Subtract 1: \( -6 \leq -2x \leq 8 \)
2
Divide by \(-2\): signs flip! \( 3 \geq x \geq -4 \)
3
Rewrite in standard order: \( -4 \leq x \leq 3 \)
-4 3

Special Case: No Solution

Sometimes an "and" compound inequality has no overlap. This happens when the two conditions contradict each other.

2 5 No overlap

Special Case: All Real Numbers

Sometimes an "or" compound inequality covers the entire number line. The two regions overlap or together span all values.

1 4 Solution: all real numbers

Common Mistakes to Avoid

✕ Forgetting to Flip
When dividing or multiplying by a negative, you must reverse the inequality sign. Missing this gives the wrong direction.
✕ Mixing Up And/Or
Writing "and" when the problem says "or" (or vice versa) changes the graph from a segment to two rays.
✕ Wrong Circle Type
Using a closed circle for \( < \) or an open circle for \( \leq \). Check the symbol before drawing.

Summary Reference

Feature "And" "Or"
Keyword Intersection Union
Graph Bounded segment Two rays
Can have no solution? Yes (no overlap) Rarely
Can be all reals? Rarely Yes (overlapping rays)
Compact notation \( a < x < b \) Two separate statements

Question 1 of 5

Solve the compound inequality: \( -1 < x + 3 < 7 \). What is the solution?

Question 2 of 5

Which graph represents \( x \leq -2 \) or \( x > 3 \)?

Question 3 of 5

Solve: \( 2x + 5 > 1 \) and \( 2x + 5 < 13 \)

Question 4 of 5

What happens when you solve \( x > 6 \) and \( x < 1 \)?

Question 5 of 5

Solve: \( -3x + 6 \geq 0 \) and \( -3x + 6 \leq 15 \)