# How do you solve the inequality 9 > -3 (x - 1) ≥ -12?

Jun 24, 2018

$- 2 < x \le 5$

#### Explanation:

Given: $9 > - 3 \left(x - 1\right) \ge - 12$

$\textcolor{b l u e}{\text{Manipulate the same way you do equation.}}$
$\textcolor{b l u e}{\text{but be mindful about the inequality sign}}$

Divide throughout by 3 (does not change the signs)

$3 > - x + 1 \ge - 4$

Subtract 1 throughout

$2 > - x \ge - 5$

$\textcolor{b r o w n}{\text{Consider the part: } - x \ge - 5}$

Swap sides and change the negatives to positives. Keep the inequality the same way round.

Same thing as: add 5 to both sides and add $x$ to both sides and keep the inequality the same way round.

Same thing as: multiply both sides by (-1) and turn the inequality sign the other way round.

$5 \ge x \textcolor{w h i t e}{\text{ddd}} \leftarrow$ This is the same thing as $x \le 5$

$\textcolor{b r o w n}{\text{Consider the part: } 2 > - x}$

Swap sides and change the signs but keep the inequality the same way round.

$x > - 2$

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$\textcolor{b r o w n}{\text{Putting it all together}}$

$- 2 < x \le 5$