Solve the inequality. Graph the solutions.

Question

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:

$$x<-30\text{or}x>0.$$

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.