# How do you solve graphically abs(x – 4)>abs(3x – 1)?

Aug 10, 2018

$- \frac{3}{2} < x < \frac{5}{4}$

#### Explanation:

The first thing you do is to draw the graphs $y = \left\mid x - 4 \right\mid$ and $y = \left\mid 3 x - 1 \right\mid$ on the SAME graph

graph{(y-abs(x-4))(y-abs(3x-1))=0 [-20.28, 20.27, -10.14, 10.13]}

Now the question is what parts of the graph above satisfies the equation $\left\mid x - 4 \right\mid > \left\mid 3 x - 1 \right\mid$. What it is asking you is what part of the graph $y = \left\mid x - 4 \right\mid$ is above the graph $y = \left\mid 3 x - 1 \right\mid$.

Hence, going from the right, the equation of each branch is $y = x - 4$, $y = 3 x - 1$, $y = 4 - x$ and $y = 1 - 3 x$.

The branches $y = 4 - x$ and $y = 3 x - 1$ meet at a point and so do $y = 4 - x$ and $y = 1 - 3 x$

Therefore, we need to find the point of intersection

$y = 4 - x$ and $y = 3 x - 1$
$4 - x = 3 x - 1$
$5 = 4 x$
$x = \frac{5}{4}$

$y = 4 - x$ and $y = 1 - 3 x$
$4 - x = 1 - 3 x$
$2 x = - 3$
$x = - \frac{3}{2}$

Finally, looking at where $y = \left\mid x - 4 \right\mid$ is above the graph $y = \left\mid 3 x - 1 \right\mid$, we can tell that it is $- \frac{3}{2} < x < \frac{5}{4}$