Question

How do you solve and graph the inequality `abs(4 – v)< 5`?

Answer 1

Let's start from a graph of a function `y=|v|`.
For non-negative `v` the absolute value of `v` is itself. So, if `v>=0`, `|v|=v` and our function is the same as `y=v`.
For negative `v` the absolute value of `v` is its opposite, `-v`. So, if `v<0`, `|v|=-v` and our function is the same as `y=-v`.
That results in this graph of `y=|v|`:
graph{|x| [-10, 10, -5, 5]}

Now let's draw a graph of `y=|4-v|`. Since `|4-v|=|v-4|`, we will draw a graph of `y=|v-4|`.
According to principles of graph transformation, a graph of `y=|v-4|` is the result of a shift to the right by `4` of a graph of a function `=|v|`. Therefore, our graph looks like this:
graph{|x-4| [-10, 10, -5, 5]}

Now, to solve `|4-v|<5`, we have to find all values of `v`, where the graph lies below the horizontal line that intersects the Y-axis at point `y=5`. Obviously, it's a segment `(-1,9)` because, while `v` is changing from `-1` to `9`, the value of `|4-v|` is changing from `5` down to zero and up to `5`, always staying below the horizontal line `y=5`.
Outside of this segment, that is if `v<=-1` or `y>=9` the value of `|4-v|` is equal or greater than `5`.

Answer 2

For those interested in purely algebraic solution, here is how to do it.
Since, by definition,
`|X|=X` if `X>=0` and
`|X|=-X` if `X<0`,
we will consider two cases.

Case 1. Seeking solutions that satisfy the inequality
`4-v>=0` or, equivalently, `v<=4`.
In this case `|4-v|=4-v` and our original inequality looks like this:
`4-v<5`.
Solution to this is `v > -1`.
Combined with the condition `v<=4` of this case, we have a segment of values of `v`, where `|4-v|<5`:
`-1 < v <= 4`

Case 2. Seeking solutions that satisfy the inequality
`4-v<0` or, equivalently, `v>4`.
In this case `|4-v|=-(4-v)` and our original inequality looks like this:
`-(4-v)<5`, that is `-4+v<5`.
Solution to this is `v < 9`.
Combined with the condition `v>4` of this case, we have another segment of values of `v`, where `|4-v|<5`:
`4 < v < 9`

Now it's appropriate to combine two segments that represent the solutions of an original inequality into one segment since these segments are adjacent:
`-1 < v <= 4` combined with `4 < v < 9` results in a segment `-1 < v < 9`
As you see, we get the same solution as using a graph above (which should not be a surprise).