Answer
$(-∞, -7) ∪ (1, ∞)$
Work Step by Step
Consider the Inequality as follows:
$|x^{2} +6x+1| > 8$
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.
$x^{2} +6x+1= 8$
This implies $x = -7, 1$
And
$x^{2} + 6x + 1= - 8$
This implies $x = -3,-3$
These solutions are used as boundary points on a number line.
2. Locate these boundary points on a number line found in Step 1 to divide the number line into intervals.
The boundary points divide the number line into four intervals:
$(-∞, -7), (-7,-3), (-3, 1) ,(1, ∞)$
3. Now, one test value within each interval is chosen and $f$ is evaluated at that number.
Intervals: $(-∞, -7) (-7,-3) (1, ∞) (-3, 1)$
Test value: $-8$ $-5$ $2$ $0$
Sign Change: Positive Negative Positive Negative
$f (x) > 0?$: T F T F
4. Write the solution set, selecting the interval or intervals that satisfy the given inequality. Solve, $f (x) > 0$, where $f (x) =x^{2} +6x – 7$.
Based on our work done in Step 4, we see that for all $x$ in$(-∞, -7)$ or $(1, ∞)$.
Conclusion: Thus, the interval notation of the given inequality is $(-∞, -7) ∪ (1, ∞)$ and the graph of the solution set on a number line is shown as follows:
