# How do you solve abs(x-5)= 5 ?

Mar 3, 2018

The solutions are $x = 0 , 10$.

#### Explanation:

There are two possible solutions to an absolute value equation.

For example, the solutions for $| x | = 1$ are $x = 1$ and $x = - 1$, because the absolute values of $1$ and $- 1$ are both $1$.

The equation $| x - 5 | = 5$ splits into to solutions: $x - 5 = 5$ and $x - 5 = - 5$. Now, solve for $x$ in both of the new equations.

$q \quad q \quad q \quad q \quad | x - 5 | = 5$
$\text{ ↙ ↘}$
color(white){color(black)( (x-5=5, qquadqquad x-5=-5), (x=10, qquadqquad x=0)):}

Those are the two solutions. We can verify them by plugging them into the original equation:

$| x - 5 | = 5$

Check $0$:

$| 0 - 5 | = 5$

$| - 5 | = 5$

5=5" "sqrt

The solution $0$ works. Check $10$:

$| 10 - 5 | = 5$

$| 5 | = 5$

5=5" "sqrt

The solution $10$ also works, so both solutions are correct. Hope this helps.

Mar 3, 2018

$x = 0 \mathmr{and} 10$
$x - 5 = 5$ or $x - 5 = - 5$
$x = 5 + 5$ or $x = - 5 + 5$
$x = 10$ or $x = 0$