# Given the function f(x)= 0.5abs(x -4)-3, for what values x is f(x)=7?

Dec 13, 2016

$x = 24$ and $x = - 16$

#### Explanation:

Given $f \left(x\right) = 0.5 \left\mid x - 4 \right\mid - 3$ and $f \left(x\right) = 7$ we can write:

$0.5 \left\mid x - 4 \right\mid - 3 = 7$

Now, we must isolate the absolute value term while keeping the equation balanced:

$0.5 \left\mid x - 4 \right\mid - 3 + 3 = 7 + 3$

$0.5 \left\mid x - 4 \right\mid - 0 = 10$

$0.5 \left\mid x - 4 \right\mid = 10$

$\frac{0.5 \left\mid x - 4 \right\mid}{0.5} = \frac{10}{0.5}$

$\frac{\cancel{0.5} \left\mid x - 4 \right\mid}{\cancel{0.5}} = 20$

$\left\mid x - 4 \right\mid = 20$

Because the absolute value function converts any number to a positive number we must solve the term within the absolute value for both 20 and -20:

$x - 4 = 20$

$x - 4 + 4 = 20 + 4$

$x - 0 = 24$

$x = 24$

and

$x - 4 = - 20$

$x - 4 + 4 = - 20 + 4$

$x - 0 = - 16$

$x = - 16$