# How do you solve y=2abs(x-4)-3?

$x \ge 4 , y = 2 x - 11$
$x \le 4 , y = - 2 x + 5$

#### Explanation:

First keep in mind that the absolute value function will only return positive values. And so we need to keep in mind that the value inside of the absolute value sign could be positive, or could be negative. We solve for both cases.

Positive

$y = 2 \left(x - 4\right) - 3$

$y = 2 x - 8 - 3$

$y = 2 x - 11$

Negative

$y = 2 \left(- x + 4\right) - 3$

$y = - 2 x + 8 - 3$

$y = - 2 x + 5$

We now have the two lines that will define the solution. However, we don't want all the points on the two lines. Let's see where they intersect to help define the points we want:

$2 x - 11 = y = - 2 x + 5$

4x=16=>color(blue)(bar( ul( abs( color(black)(x=4))))

:. y=2(4)-11=>color(blue)(bar( ul( abs( color(black)(y=-3))))

And so:

$x \ge 4 , y = 2 x - 11$
$x \le 4 , y = - 2 x + 5$

The graph of this solution looks like this:

graph{2abs(x-4)-3}