Question

How do you solve and graph `abs(2c-1)<=7`?

Answer 1

`-3 \ le c \ le 4`

Explanation:

`| 2c-1 | \le7`

Consider a general inequality with the absolute value:
`| f(x) | \le a`

By definition this is equivalent to solving the following inequalities:
`f (x) \ le a` and `f (x) \ ge -a`

In our specific case this means:
`2c-1 \ le 7`
`2c-1 \ ge -7`

So let's solve the first one:
`2c-1 \ le 7`
`\ color (blue) {(1)} + 2c -1 \ le 7+ \ color (blue) {(1)}`
`2c \ le 8`
`\ frac {2} {2} c \ le \ frac {8} {2}`
`c \ le 4`

Now the second:
`2c-1 \ ge -7`
`2c \ ge -6`
`c \ ge -3`

This is the graph of the first inequality:
graph{x <= 4 [-6.244, 6.243, -3.12, 3.123]}

And this of the second:
graph{x => - 3 [-4.163, 1.997, -1.455, 1.624]}

So the solution and the final graph are the common part of the two inequalities:
`-3 \ le c \ le 4`