# How do you solve the absolute value equation x+absx=28?

Apr 1, 2015

If $x < 0$, the $\left\mid x \right\mid = - x$ and $x + \left\mid x \right\mid = 0 \ne 28$ so we must have $x > 0$ and $\left\mid x \right\mid = x$ so we need $x + x = 28$ and $x = 14$

Apr 1, 2015

Answer is $x = 14$

To solve this equation you need to understand what $| x |$
means.
The Magnitude of $x$ (denoted |x|) simply means the positive side of a number($x$)

If $x$ is positive or $0$ ($x \ge 0$) then $| x | = x$
and if $x$ is negative ($x < 0$) it imples $| x | = - x$

Example:
$| 5 | = 5$ because $5$ is already positive
$| - 11 | = - \left(- 11\right)$ because $- 11$ is negative

Back to our question:
$x + | x | = 28$

There will be at most two values of $x$, because there are two possibilities; either $x \ge 0$ or $x < 0$

Case 1: $x \ge 0$
$\implies x + x = 28$

$\implies 2 x = 28 \implies x = 14$

Case 2: $x < 0$
$\implies x - x = 28 \implies 0 = 28$
but we know that $0 \ne 28$ So the above(last) statement is inconclusive

So the only value we can retain is $x = 14$