Question

How do you solve and write the following in interval notation: `|-4x| + |-5| <=9`?

Answer 1

Solve for `|x|` and find the bounds for which `x` must lie in.

Explanation:

Since constants can be taken out of absolute values are their positive values when alone or when multiplied with a variable (but not when adding/subtracting), we can rewrite the inequality as:
`4|x|+5<=9`

Now we solve for `|x|`:
`|x|<=1`

We can remove the absolute value by recognizing that `x` must be equal to or between `-1` and `1`:
`-1<=x<=1`

In interval notation, parentheses are used for greater than (>), less than (<), and infinity (both `-oo` and `oo`). Brackets are used for greater than or equal to (`>=`) and less than or equal to (`<=`).
Therefore, `-1<=x<=1` can be written as `[-1,1]` in interval notation.