Question

How do you solve the inequality `2abs(7-x)+4>1` and write your answer in interval notation?

Answer 1

`[11/2,17/2]`

Explanation:

Isolate the absolute value (in red):
`2\color{red}{|7-x|}+4\gt1`
`2\color{red}{|7-x|}\gt1-4`
`\color{red}{|7-x|}\gt\frac{-3}{2}`

---

Expand absolute value:
`-3/2\lt7-x\lt-(-3/2)`
`-3/2\lt7-x\lt3/2`
`-3/2-7\lt-x\lt3/2-7`

Note the sign change here. This happens when you divide by negatives.
`3/2+7\color{tomato}{\gt}x` and `x\color{tomato}{\gt}-3/2+7`
Effectively, this means `-3/2+7\ltx\lt3/2+7`

Simplifying new inequality:
`-3/2+7\ltx\lt3/2+7`
`-3/2+14/2\ltx\lt3/2+14/2`
`11/2\ltx\lt17/2`

---

In interval notation:
`[11/2,17/2]` (brackets are used when you have `\lt,\gt` signs).

Answer 2

The solution is `x in (-oo,+oo)`

Explanation:

Let

`7-x=0`

`=>`, `x=7`

`AA x in RR`

`2|7-x|>0`

And

`2|7-x|+4>1`

The solution is

`x in (-oo,+oo)`

graph{2|7-x|+3 [-21.99, 29.33, -11.31, 14.36]}