Linear Inequalities

Solve $x - 2 \leq 7$ . Graph the solution set. Use interval notation to write the solution set.

$x - 2 \leq 7$

Add 2 to each side.

$x - 2 + 2 \leq 7 + 2$

$x \leq 9$

or

The interval notation for the solution set is $(- \infty, 9]$ .

Solve $3 x > - 6$ . Graph the solution set. Use interval notation to write the solution set.

$3 x > - 6$

Divide each side by 3. Because 3 is positive, do not reverse the inequality symbol's direction.

$\frac{3 x}{3} > \frac{- 6}{3}$

$x > - 2$

or

The interval notation for the solution set is $(- 2, \infty)$ .

Solve $- 2 x \geq 8$ . Graph the solution set. Use interval notation to write the solution set.

$- 2 x \geq 8$

Divide each side by $- 2$ . Because $- 2$ is negative, reverse the inequality symbol's direction.

$\frac{- 2 x}{- 2} \leq \frac{8}{- 2}$

$x \leq - 4$

or

The interval notation for the solution set is $(- \infty, - 4]$ .

Solve $- 10 \leq 3 x + 2 < 5$ . Graph the solution set. Use interval notation to write the solution set.

$- 10 \leq 3 x + 2 < 5$

Subtract 2 from all three parts.

$- 10 - 2 \leq 3 x + 2 - 2 < 5 - 2$

$- 12 \leq 3 x < 3$

Divide each side by 3. Because 3 is positive, do not reverse the inequality symbol's direction.

$\frac{- 12}{3} \leq \frac{3 x}{3} < \frac{3}{3}$

$- 4 \leq x < 1$

or

The interval notation for the solution set is $[- 4, 1)$ .

Solve $-4 < - \frac{2}{3} x \leq 6$ . Graph the solution set. Use interval notation to write the solution set.

$-4 < - \frac{2}{3} x \leq 6$

Multiply each part by $- \frac{3}{2}$ . Since $- \frac{3}{2}$ is negative, reverse both inequality symbols' directions.

$(- \frac{3}{2}) (-4) > (- \frac{3}{2}) (- \frac{2}{3} x) \geq (- \frac{3}{2}) (6)$

$6 > x \geq - 9$

"Turn around" the entire inequality $6 > x \geq - 9$ to rewrite it as $- 9 \leq x < 6$ .

$- 9 \leq x < 6$

or

The interval notation for the solution set is $[- 9, 6)$ .