# How do you solve and write the following in interval notation:  abs(x-5)<=7?

Jul 12, 2016

$- 2 \le x \le 12.$

In the Interval Notation, we write this as, $x \in \left[- 2 , 12\right] .$

#### Explanation:

Recall the Defn. of the Absolute Value of t in RR : |t|=t, if t>=0, &, |t|=-t, if t<0.

So, we need to consider two cases $: \left(i\right) \left(x - 5\right) \ge 0 , \left(i i\right) \left(x - 5\right) < 0$

Case $\left(i\right) : \left(x - 5\right) \ge 0.$
$\therefore | x - 5 | = x - 5$, & hence,
$| x - 5 | \le 7 \Rightarrow x - 5 \le 7 \Rightarrow x \le 5 + 7 = 12.$

Case $\left(i i\right) : \left(x - 5\right) < 0.$
$\therefore | x - 5 | = - \left(x - 5\right) = 5 - x$, so that,
$| x - 5 | \le 7 \Rightarrow 5 - x \le 7 \Rightarrow - 2 \le x .$

Combining these cases, we have, $- 2 \le x \le 12.$

In the Interval Notation, we write this as, $x \in \left[- 2 , 12\right] .$