# How do you solve |x|=-5x+24 and find any extraneous solutions?

Oct 31, 2017

The piecewise definition of |x| = {(x;x>=0),(-x;x<0):} allows one to separate the equation into two equations, one with x and the other with -x and then solve both equations.
Check.

#### Explanation:

Separate $| x | = - 5 x + 24$ into two equations:

x = -5x+24; x >=0 and -x = -5x+24; x < 0

Add 5x to both sides of both equations:

6x = 24; x>=0 and 4x = 24;x<0

Divide the first equation by 6 and the second equation by 4:

x = 4; x>=0 and x = 6;x<0

We must discard the $x = 6$ solution, because it is not within the domain restriction.

If we had not included the domain restrictions, we would have discovered the extraneous root by performing the check.

Check:

$| 4 | = - 5 \left(4\right) + 24$ and $| 6 | = - 5 \left(6\right) + 24$

$| 4 | = 4$ and $| 6 | \ne - 6$

Only the $x = 4$ solution checks, therefore, we must discard the $x = 6$ solution.