Question

How do you graph and solve `|3-2x|>7`?

Answer 1

Solution: `color(blue)(x<-2 or x >5`

Using Interval Notation: `color(blue)((-oo, -2) uu (5, oo))`

Explanation:

Given the Inequity Expression:

`color(red)(|3-2x| > 7)`

We must remember that,

`color(blue)(|f(x)| > a rArr f(x) < -a or f(x) > a)`

Hence,

we consider two possibilities as shown below:

`color(green)(3-2x<-7 or 3-2x >7)`

First, we will consider

`color(red)(3-2x < -7`

Subtract 3 from both sides

`rArr3-2x-3 < -7-3`

`rArr cancel 3-2x-cancel 3 < -7-3`

Simplify to get

`rArr -2x < -10`

Multiply both the sides by `(-1)`

We must remember to reverse the inequality

`rArr -2x (-1) > -10(-1)`

Simplify to get

`rArr 2x > 10`

Divide both sides by the value of `(2)`

`rArr (2x)/2 > 10/2`

`rArr (cancel 2x)/cancel 2 > 10/2`

`rArr x> 5` Intermediate Result 1

Next, we will consider

`color(red)(3-2x >7`

Subtract the value of 3 from both sides

`3-2x - 3 >7 - 3`

`rArr cancel 3-2x - cancel3 >7 - 3`

`rArr -2x > 4`

Multiply both the sides by `(-1)`

We must remember to reverse the inequality

`rArr -2x (-1) < 4 (-1)`

`rArr 2x < -4`

Divide both sides by the value of 2

`rArr (2x)/2 < -4/2`

`rArr (cancel 2x)/cancel 2 < -4/2`

`rArr x < -2` Intermediate Result 2

Let us now combine both the intermediate results (1) & (2) to get

`color(blue)(x < -2 or x > 5)`

Using the Interval Notation we can can also write the solution as

`color(green)((-oo, -2) uu (5, oo))`

Please refer to the inequality graph below which demonstrates our solution: