Question

Solve the inequality. Graph the solutions.
StartAbsoluteValue StartFraction x plus 15 Over 5 EndFraction EndAbsoluteValue
x+15
5greater than>33
Question content area bottom
Part 1
Select the correct choice below and fill in any answer boxes in your choice.
A.
The solution is xequals=enter your response here. ​(Type an integer or a simplified​ fraction.)
B.
The solution is enter your response hereless than<xless than<enter your response here. ​(Type integers or simplified​ fractions.)
C.
The solution is xless than<enter your response here or xgreater than>enter your response here. ​(Type integers or simplified​ fractions.)
D.
The solution is all real numbers.
E.
There is no solution.

Answer 1

To solve the inequality

\[
\left| \frac{x + 15}{5} \right| > 33,
\]

we'll start by manipulating the absolute value inequality into two separate inequalities.

1. We can start by setting the fraction inside the absolute value greater than 33:

\[
\frac{x + 15}{5} > 33
\]

2. Next, we also need to consider the case when the fraction inside the absolute value is less than -33:

\[
\frac{x + 15}{5} < -33
\]

Now we will solve each of these inequalities separately.

**For the first inequality:**

\[
\frac{x + 15}{5} > 33
\]

Multiply both sides by 5 (since 5 is positive, the direction of the inequality remains the same):

\[
x + 15 > 165
\]

Subtract 15 from both sides:

\[
x > 150
\]

**For the second inequality:**

\[
\frac{x + 15}{5} < -33
\]

Again, multiplying both sides by 5:

\[
x + 15 < -165
\]

Subtract 15 from both sides:

\[
x < -180
\]

Now we can summarize the solutions. The solution set for the original inequality is:

\[
x < -180 \quad \text{or} \quad x > 150.
\]

Thus, the answer corresponds to Choice C:

**C. The solution is** \(x < -180\) **or** \(x > 150\).

### Graphing the Solutions

The graph of the solutions will consist of two open intervals on a number line:

- An open interval extending left from -180.
- An open interval extending right from 150.

So the number line would look like this:

```
<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--->
-200 -180 -150 -100 -50 0 50 100 150 200
(------) (-------)
```

This represents \( x < -180 \) or \( x > 150 \).