How do you graph and solve 3|8-x| + 2 <7- 2|x-8|?

Jan 4, 2016

$7 \le x \le 9$

The graph is all the area between and including x =7 ; x=9

Explanation:

Given: $3 | 8 - x | + 2 < 7 - 2 | x - 8 |$

Collecting like terms

$3 | 8 - x | + 2 | x - 8 | < 7 - 2$

But $| 8 - x | = | x - 8 | \to t e s t \to | 8 - 2 | = | 2 - 8 | \to 6 = 6$

using only $| 8 - x |$ for both

Factoring out gives:

$| 8 - x | \left(3 + 2\right) < 5$

$| 8 - x | < 1$

Absolute is always 'not negative' so we need:

$0 \le | 8 - x | < 1$

If $x = 9 \text{ then } | 8 - x | = 1$
If $x > 9 \text{ then } | 8 - x | > 1$
If $x = 7 \text{ then } | 8 - x | = 1$
If $x < 7 \text{ then } | 8 - x | > 1$

So $\textcolor{w h i t e}{.} 7 \le x \le 9$