# How do you solve abs(x + 1)=abs(-4 x + 3)?

May 25, 2018

The solutions are $\left\{\begin{matrix}x = \frac{4}{3} \\ x = \frac{2}{5}\end{matrix}\right.$

#### Explanation:

The equation is

$| x + 1 | = | - 4 x + 3 |$

Removing one absolute value at a time :

$| x + 1 | = - 4 x + 3$ and $| x + 1 | = 4 x - 3$

The first equation gives

$| x + 1 | = - 4 x + 3$

$\iff$, $\left\{\begin{matrix}x + 1 = - 4 x + 3 \\ - x - 1 = - 4 x + 3\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}5 x = 2 \\ 3 x = 4\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}x = \frac{2}{5} \\ x = \frac{4}{3}\end{matrix}\right.$

The second equation gives

$| x + 1 | = 4 x - 3$

$\iff$, $\left\{\begin{matrix}x + 1 = 4 x - 3 \\ - x - 1 = 4 x - 3\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}3 x = 4 \\ 5 x = 2\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}x = \frac{4}{3} \\ x = \frac{2}{5}\end{matrix}\right.$

graph{(y-|x+1|)(y-|-4x+3|)=0 [-6.38, 7.67, -1.74, 5.283]}