# How do you graph and solve abs[ x - 3 ] <=5 ?

Jun 10, 2018

$\left\{x | x \setminus \in \left[- 2 , 8\right] , x \setminus \in m a t h \boldsymbol{R}\right\}$

#### Explanation:

for $x \ge q 3 , x - 3 \ge q 0$ thus for $x \ge q 3 , | x - 3 | = x - 3$
for $x \le q 3 , x - 3 \le q 0$ thus for $x \le q 3 , | x - 3 | = 3 - x$

Solve for the two cases:
for $x \ge q 3 , x - 3 \setminus \le q 5 ,$
$x \setminus \le q 8$
thus $3 \setminus \le q x \le q 8$

for $x \le q 3 , 3 - x \setminus \le q 5 ,$
$- 2 \setminus \le q x$
thus $- 2 \setminus \le q x \le q 3$

Find the union of these two intervals:
$\left[- 2 , 3\right] \setminus \cup \left[3 , 8\right] = \left[- 2 , 8\right]$

Thus the solution is $\left\{x | x \setminus \in \left[- 2 , 8\right] , x \setminus \in m a t h \boldsymbol{R}\right\}$

Graph:

Jun 10, 2018

$- 2 \le x \le 8$

#### Explanation:

$| x - 3 | < = 5$
The simplest way to solve this type of inequality is solving it in 2 separate steps:
a. $\left(x - 3\right) \le 5$ --> $x \le 8$
b. $- \left(x - 3\right) \le 5$ --> $- x \le 2$
$x \ge - 2$
Answer: $- 2 \le x \le 8$