Compound inequalities

$- 2 < x \leq 3$ is really two statements glued together: $x > - 2$ and $x \leq 3$ . Its solutions are the numbers between them.

- 2 < x \leq 3

lives between the two marked points.

Solve a three-part inequality by doing the same thing to all three parts.

Work through it

Subtract 3 from all three parts. $- 2 < 2 x \leq 6$

Divide all three by 2. $- 1 < x \leq 3$

Solve each side separately. $x < - 1$ or $x > 4$

Or keeps both pieces, so the answer is two separate rays. $x < - 1$ or $x > 4$

Writing $5 < x < 2$ .

No number is above 5 and below 2. If that happens in a solution, check your signs, since it might really be or.

The short version
And means both must be true, so look for the overlap. Or means at least one must be true.

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