Answer
$(-∞, -8) ∪ (-6, 4) ∪ (6, ∞) $
Work Step by Step
Consider the Inequality as follows:
$|x^{2} -2x -36| > 12$
Here are the steps required for Solving Modulus Inequalities:
1. One side must be zero and the other side can have only one fraction, so we simplify the fractions if there is more than one fraction.
$x^{2} -2x -36 = 12$
This implies
$x = -8, 6$
and
$x^{2} -2x -36 = -12$
This implies
$x = -6, 4$
These solutions are used as boundary points on a number line.
2. Locate these boundary points on a number line found in Step 2 to divide the number line into intervals.
The boundary points divide the number line into five intervals:
$(-∞, -8), (-8, -6), (-6, 4), (4, 6), (6, ∞)$
3. Now, one test value within each interval is chosen and f is evaluated at that number.
Intervals:$ (-∞, -8) (-8,-6) (-6, 4) (4, 6) (6, ∞)$
Test value: $9$ $ -5$ $ 0$ $5$ $7 $
Sign Change: Positive Negative Positive Negative Positive
$f (x) > 0?$: T F T F T
4. Write the solution set, selecting the interval or intervals that satisfy the given inequality.
Solve, $f (x) > 0$, where $f (x) = x^{2}-2x – 48$. Based on our work done in Step 4, we see that for all x in $(-∞, -8)$ or $(-6, 4)$ or $(6, ∞)$.
Thus, we conclude that the interval notation of the given inequality is
$(-∞, -8) ∪ (-6, 4) ∪ (6, ∞)$.
The graph of the solution set on a number line is shown as follows:
