# How do you solve abs(x+3)=abs(7-x)?

Mar 30, 2015

One way(Method 1) would be to replace each of the absolute value expressions with 2 possible expressions (one positive and one negative). For the given example there are two absolute value expression and all 4 possible combinations would need to be considered.

An alternative (Method 2) would be to square both sides and solve.

Method 1
$- \left(x + 3\right) = - \left(7 - x\right)$
$\rightarrow - 2 x = - 4$
$\rightarrow x = 2$

$- \left(x + 3\right) = + \left(7 - x\right)$
$\rightarrow - 3 = 7$ impossible; extraneous result; ignore

$+ \left(x + 3\right) = - \left(7 - x\right)$
$\rightarrow 3 = - 7$ Impossible; extraneous result; ignore

$+ \left(x + 3\right) = + \left(7 - x\right)$
$\rightarrow 2 x = 4$
$\rightarrow x = 2$ (a duplicate of the second result)

The only solution is $x = 2$

Method 2
${\left(\left\mid x + 3 \right\mid\right)}^{2} = {\left(\left\mid 7 - x \right\mid\right)}^{2}$

$\cancel{{x}^{2}} + 6 x + 9 = 49 - 14 x + \cancel{{x}^{2}}$
$\rightarrow 20 x = 40$
$x = 2$