Ask a New Question
Menu

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.

1 answer

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:

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

  1. 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 \( x > 150 \).