# How do you solve |-4x +7| = x + 17?

Mar 13, 2016

$x = - 2 , 8$

#### Explanation:

$1$. Recall that the absolute value equality property, $| a | = b$, can be written as $a = \pm b$. Thus, there will be two solutions.

$| - 4 x + 7 | = x + 17 \textcolor{w h i t e}{X} , \textcolor{w h i t e}{X} \text{gives:}$

$- 4 x + 7 = x + 17 \textcolor{w h i t e}{X X} \textcolor{p u r p \le}{\text{or}} \textcolor{w h i t e}{X X} - 4 x + 7 = - \left(x + 17\right)$

$2$. For each equation, solve for $x$.

$\textcolor{w h i t e}{X X X x} - 5 x = 10 \textcolor{w h i t e}{X X} \textcolor{p u r p \le}{\text{or}} \textcolor{w h i t e}{X X} - 4 x + 7 = - x - 17$

$\textcolor{w h i t e}{X X X X X} x = \frac{10}{-} 5 \textcolor{w h i t e}{X X X X x} - 3 x = - 24$

$\textcolor{w h i t e}{X X X x} \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} x = - 2 \textcolor{w h i t e}{\frac{a}{a}} |}}} \textcolor{w h i t e}{X X X x} x = \frac{- 24}{- 3}$

$\textcolor{w h i t e}{X X X X X X X X X X X X \times} \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} x = 8 \textcolor{w h i t e}{\frac{a}{a}} |}}}$

If you graph the equation, you can see that the intersection points occur when $x = - 2$ and $x = 8$:

$\therefore$, $x = - 2$ or $8$.