Question

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

Answer 1

`{ x | x \in [-2,8], x \in mathbb{R} }`

Explanation:

for `x geq 3 , x-3 geq0` thus for `x geq 3 , |x-3| = x - 3`
for `x leq 3 , x-3 leq0` thus for `x leq 3 , |x-3| = 3-x`

Solve for the two cases:
for `x geq 3, x - 3 \leq 5,`
`x\leq 8`
thus `3 \leq x leq 8`

for `x leq 3, 3-x \leq 5,`
`-2\leq x`
thus `-2 \leq x leq 3`

Find the union of these two intervals:
`[-2,3] \cup [3,8] = [-2,8]`

Thus the solution is `{ x | x \in [-2,8], x \in mathbb{R} }`

Graph:

Answer 2

`-2 <= x <= 8`

Explanation:

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