How do you solve -1 <= x-3 <= 5?

Mar 16, 2017

Solve for $x$ by adding $+ 3$ to all three parts of the compound the inequality.

$2 \le x \le 8$

Explanation:

$- 1 \le x - 3 < - 5 \text{ }$add 3 to each part

$- 1 \textcolor{red}{+ 3} \le x - 3 \textcolor{red}{+ 3} \le 5 \textcolor{red}{+ 3} \text{ }$ Doing all of the addition results in

$+ 2 \le x \le 8 \text{ } x$ is now isolated in the middle

$2 \le x \le 8 \text{ }$ this is the final answer

Mar 16, 2017

$2 \le x \le 8$

Explanation:

$- 1 \le x - 3 \le 5$

You can break the inequality up into two parts, work with them separately, and combine them at the end.

$\textcolor{red}{- 1 \le x - 3} \le 5 \text{ "and" } - 1 \le \textcolor{b l u e}{x - 3 \le 5}$

$\textcolor{red}{- 1 \le x - 3} \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} \textcolor{b l u e}{x - 3 \le 5}$

Isolate x on the right side $\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots .}$ isolate x on the left side
$\textcolor{red}{- 1 + 3 \le x} \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} \textcolor{b l u e}{x \le 5 + 3}$

$\textcolor{w h i t e}{\ldots \ldots \ldots} \textcolor{red}{2 \le x} \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . .} \textcolor{b l u e}{x \le 8}$

Now combine the two parts with one $x$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} \textcolor{red}{2 \le} x \textcolor{b l u e}{\le 8}$