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

1 Answer
Jan 12, 2018

Answer:

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:

enter image source here