Question

How do you solve `4| v + 7| + 8> - 20`?

Answer 1

See below.

Explanation:

To solve inequalities that involve an absolute value, we need to account for both positive and negative values in the absolute bars.

So we have:
`color(blue)([1])->`4|v + 7| + 8 > -20#

And:
`color(blue)([2])->`4|-(v + 7)| + 8 > -20#

For `color(blue)([1])` remove bars:

`4|v + 7| + 8 > -20=>` `4(v + 7) + 8 > -20`

Divide both sides by `4`:

`v + 7 +2 > -5`

Subtract `9` from both sides:

`color(blue)(v > -14)`

For `color(blue)([2])` remove bars:

`4|-(v + 7)| + 8 > -20=>` `-4(v + 7) + 8 > -20`

Divide both sides by `-4` remembering to reverse the inequality sign.

`(v + 7) -2 < 5`

Add `2` to both sides and remove bracket:

`v + 7 < 7`

Subtract `7` from both sides:

`color(blue)(v < 0)`

Solutions in interval notation:

`( -14 , 0 )`

`color(red)(WARNING)`

This is a general method for solving inequalities with absolute values. Although the solution interval is a valid solution, it is not the true solution to this inequality. On observation it can be seen that the minimum possible value of the `| v + 7|` is `0`:

This gives:

`0 + 8 > -20`

This will be true for all real `v`.

So the true solution is: `( -oo , oo)`

This type of error can often occur in inequalities, so it is always best to check for this.