Question

|x-3| < 5 please help

Answer 1

`\left|x-3\right|<5\quad :\quad \begin{bmatrix}\mathrm{Solution:}\:&\:-2<x<8\:\\ \:\mathrm{Interval\:Notation:}&\:\left(-2,\:8\right)\end{bmatrix}`

The graph is also attached.

Explanation:

Given the expression

`|x-3|\:<\:5`

Apply absolute rule:

`\mathrm{If}\:|u|\:<\:a,\:a>0\:\mathrm{then}\:-a\:<\:u\:<\:a`

so the expression becomes

`-5<x-3<5`

`x-3>-5\quad \mathrm{and}\quad \:x-3<5`

solving condition 1

x−3<5

Add 3 to both sides

x−3+3<5+3

x<8

solving condition 2

x−3>−5

Add 3 to both sides

x−3+3>−5+3

x>−2

combining the intervals

`x>-2\quad \mathrm{and}\quad \:x<8`

Merging overlapping intervals

`-2<x<8`

Therefore,

`\left|x-3\right|<5\quad :\quad \begin{bmatrix}\mathrm{Solution:}\:&\:-2<x<8\:\\ \:\mathrm{Interval\:Notation:}&\:\left(-2,\:8\right)\end{bmatrix}`

The graph is also attached.