Use algebraic​ and/or graphical methods to solve the inequality.

Question

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 \((0, \frac{3}{2})\).