Compound inequalities

Two conditions at once

A compound inequality joins two inequalities with "and" or "or." The small word makes a big difference.

"And": both must be true

Test some numbers for "$x > -2$ and $x \leq 3$." A number only counts if it passes both tests.

$x > -2 \text{ and } x \leq 3$

Tap a number on the line to test it.

Test every pointShow the solution set

The solutions are caught between $-2$ and $3$: where the graph of $x > -2$ and the graph of $x \leq 3$ overlap. Because $x$ is trapped in the middle, it's usually written as one three-part inequality:

$-2 < x \leq 3$

"Or": at least one must be true

Now test "$x < -1$ or $x \geq 2$." This time, passing either test is enough.

$x < -1 \text{ or } x \geq 2$

Tap a number on the line to test it.

Test every pointShow the solution set

The graph has two separate pieces heading in opposite directions. The solutions are everything in either graph, and the gap in the middle is the numbers that fail both tests.

Solving a three-part inequality

Do the same thing to all three parts to get $x$ alone in the middle.

Check the answer by testing points in the original:

$-1 \leq 2x + 3 < 9$

Tap a number on the line to test it.

Test every pointShow the solution set

If you divide by a negative, flip both signs. Then rewrite it so the smaller number is on the left.

Solving an "or" inequality

Solve each part separately, then join the answers with "or."

Key ideas

• "And" means both conditions hold: the solution is where the graphs overlap.
• "Or" means at least one condition holds: the solution combines both graphs.
• Solve a three-part inequality by doing the same thing to all three parts, flipping both signs for a negative.