How to Solve Compound Inequalities (AND and OR)

Subtract from all three parts at once.

$2 < x < 7$

Subtract $2$ from every part: $2 < x < 7$.

Check: $x = 5$ gives $4 < 7 < 9$ ✓

$-1 \le x \le 4$

Add $1$ to all three parts: $-2 \le 2x \le 8$. Divide all three by $2$: $-1 \le x \le 4$.

Check: $x = 0$ gives $-3 \le -1 \le 7$ ✓

$3 < x < 6$

First part: $x > 3$. Second part: $x < 6$. The overlap is $3 < x < 6$.

This is already solved. Describe the graph.

Two pieces: everything left of $-3$, and everything from $2$ rightwards.

Open circle at $-3$ with an arrow left; closed circle at $2$ with an arrow right. Nothing between $-3$ and $2$ is shaded, because a number there satisfies neither part.

$x < -3$ or $x \ge 4$

First: $2x < -6$, so $x < -3$. Second: $x \ge 4$.

Joined with OR, the solution is both stretches: $x < -3$ or $x \ge 4$.

No solution.

The two stretches never overlap, so no number satisfies both conditions at once.

Test a number from well outside the range $0$ to $10$.

All real numbers.

Test $x = -50$: it fails the first part but satisfies $x \le 10$, so it qualifies through OR. Test $x = 500$: it satisfies $x \ge 0$.

Every number satisfies at least one part, so the solution is all real numbers.

“At least” and “no more than” both include their boundary value.

$42 \le h \le 76$, and yes.

Both conditions must hold at once, so this is an AND, written $42 \le h \le 76$.

Both symbols include equality, so a rider of exactly $76$ inches satisfies $76 \le 76$ and can ride.