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

May 9, 2017

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:
$4 | x | + 5 \le 9$

Now we solve for $| x |$:
$| x | \le 1$

We can remove the absolute value by recognizing that $x$ must be equal to or between $- 1$ and $1$:
$- 1 \le x \le 1$

In interval notation, parentheses are used for greater than (>), less than (<), and infinity (both $- \infty$ and $\infty$). Brackets are used for greater than or equal to ($\ge$) and less than or equal to ($\le$).
Therefore, $- 1 \le x \le 1$ can be written as $\left[- 1 , 1\right]$ in interval notation.