Question

Solve the inequality. Graph the solutions.
StartAbsoluteValue StartFraction x plus 15 Over 5 EndFraction EndAbsoluteValue
x+15
5greater than>3
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

\[
\left| \frac{x + 15}{5} \right| > 3,
\]

we start by breaking it down into two separate cases, based on the definition of absolute value.

### Case 1:
\[
\frac{x + 15}{5} > 3
\]

Multiply both sides of the inequality by 5 (since 5 is a positive number, the direction of the inequality does not change):

\[
x + 15 > 15.
\]

Subtract 15 from both sides:

\[
x > 0.
\]

### Case 2:
\[
\frac{x + 15}{5} < -3
\]

Again, multiply both sides by 5:

\[
x + 15 < -15.
\]

Subtract 15 from both sides:

\[
x < -30.
\]

### Combining the Results:
From both cases, we find:

1. \( x > 0 \)
2. \( x < -30 \)

Thus, the solution to the inequality is:

\[
x < -30 \quad \text{or} \quad x > 0.
\]

### Answer Selection:
The correct choice based on the results is:

C. The solution is \(x < -30\) or \(x > 0\).

### Graph the Solutions:
To graph the solutions:
- Draw a number line.
- Shade to the left of -30 to represent \(x < -30\).
- Shade to the right of 0 to represent \(x > 0\).

This visual representation shows that the solutions include all values less than -30 and all values greater than 0.