# How do you solve -2<=x-7<=11?

Aug 3, 2016

$x \in \left[5 , 18\right]$

#### Explanation:

The first thing to do here is get $x$ alone in the middle of the compound inequality by adding $7$ to all three sides

$- 2 + 7 \le x - \textcolor{red}{\cancel{\textcolor{b l a c k}{7}}} \le 11 + 7$

$\textcolor{w h i t e}{a a a a a} 5 \le \textcolor{w h i t e}{a a} x \textcolor{w h i t e}{a a} \le 18$

You now know that in order to be part of the solution interval, a value of $x$ must satisfy two conditions

$x \ge \textcolor{w h i t e}{1} 5 \to$ the left side of the compound inequality

$x \le 18 \to$ the right side of the compound inequality

For the first condition, you need $x$ to be greater than or equal to $5$. In interval notation, this is written as

$x \in \left[5 , + \infty\right)$

For the second condition, you need $x$ to be smaller than or equal to $18$. In interval notation, this is written as

$x \in \left(- \infty , 18\right]$

This means that the solution interval for the compound inequality must have $x$ greater than or equal to $5$ and smaller than or equal to $18$.

Thsi is written as

$x \in \left(- \infty , 18\right] \cap \left[5 , + \infty\right) \implies \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{x \in \left[5 , 18\right]} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
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