# How do you solve | x - 3 | = x + 1 graphically?

Jan 13, 2018

See explanation.

#### Explanation:

Look at the equation as two functions: $y = \left\mid x - 3 \right\mid$ and $y = x + 1$.

To graph $y = \left\mid x - 3 \right\mid$ we know the the vertex is at $\left(3 , 0\right)$. The slope to the right of the vertex is 1 and the slope to the right of the vertex is $- 1$. The graph is piecewise linear. So graph $y = - \left(x - 3\right)$ for $x \le 3$ and $y = x - 3$ for $x > 3$.

The graph of $y = x + 1$ is a linear function with a $y$-intercept of $\left(0 , 1\right)$ and $x$-intercept of $\left(- 1 , 0\right)$.

If you graph carefully enough you can see that the only intersection point is at $\left(1 , 2\right)$, where the left branch of the absolute value, $y = - \left(x - 3\right)$ intersects the linear function, $y = x + 1$.

Here's the graph.
graph{(y-abs(x-3))(y-x-1)=0 [-7.83, 12.17, -2.6, 7.4]}