Question

Use algebraic​ and/or graphical methods to solve the inequality.
StartFraction 15 minus 5 x Over x EndFraction
greater than5  
Question content area bottom
Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The​ solution(s) is/are xequals
  
enter your response here.
​(Use a comma to separate answers as​ needed.)
B.
The solutions are the values of x in the interval
  
enter your response here.
​(Type your answer in interval​ notation.)
C.
There is no solution.

Answer 1

To solve the inequality

\[
\frac{15 - 5x}{x} > 5,
\]

we can first rewrite the inequality in a more workable form:

1. **Subtract 5 from both sides:**

\[
\frac{15 - 5x}{x} - 5 > 0.
\]

To combine the fractions, we rewrite 5 as \(\frac{5x}{x}\):

\[
\frac{15 - 5x - 5x}{x} > 0 \implies \frac{15 - 10x}{x} > 0.
\]

2. **Set the numerator and denominator:**

From \(\frac{15 - 10x}{x} > 0\), we can identify critical points where the expression could be equal to zero or undefined:

- The numerator \(15 - 10x = 0 \Rightarrow 10x = 15 \Rightarrow x = \frac{3}{2}\).
- The denominator \(x = 0\) makes the expression undefined.

So our critical points are \(x = 0\) and \(x = \frac{3}{2}\).

3. **Create test intervals:**

We will test intervals based on these critical points:

- \( (-\infty, 0) \)
- \( (0, \frac{3}{2}) \)
- \( (\frac{3}{2}, \infty) \)

4. **Test values in each interval:**

- **For \(x < 0\)** (e.g., \(x = -1\)):
\[
\frac{15 - 10(-1)}{-1} = \frac{15 + 10}{-1} = \frac{25}{-1} < 0.
\]

- **For \(0 < x < \frac{3}{2}\)** (e.g., \(x = 1\)):
\[
\frac{15 - 10(1)}{1} = \frac{15 - 10}{1} = 5 > 0.
\]

- **For \(x > \frac{3}{2}\)** (e.g., \(x = 2\)):
\[
\frac{15 - 10(2)}{2} = \frac{15 - 20}{2} = \frac{-5}{2} < 0.
\]

5. **Summarizing the results:**

The expression is positive in the interval \( (0, \frac{3}{2}) \) and negative outside of it (in both directions).

Since we want the values where the expression is greater than \(0\), we have:

\[
x \in (0, \frac{3}{2}).
\]

So the answer will be:

**Part B: The solutions are the values of \(x\) in the interval \((0, \frac{3}{2})\).**