Question

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

Answer 1

`-3/2 < x < 5/4`

Explanation:

The first thing you do is to draw the graphs `y=abs(x-4)` and `y=abs(3x-1)` 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 `abs(x-4) > abs (3x-1)`. What it is asking you is what part of the graph `y=abs(x-4)` is above the graph `y=abs(3x-1)`.

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

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

Therefore, we need to find the point of intersection

`y=4-x` and `y=3x-1`
`4-x=3x-1`
`5=4x`
`x=5/4`

`y=4-x` and `y=1-3x`
`4-x=1-3x`
`2x=-3`
`x=-3/2`

Finally, looking at where `y=abs(x-4)` is above the graph `y=abs(3x-1)`, we can tell that it is `-3/2 < x < 5/4`