MHB Solve the inequality and graph the solution on a real number line

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To solve the inequality (3x - 5)/(x - 5) > 4, first subtract 4 from both sides to obtain (3x - 5)/(x - 5) - 4 > 0. Next, combine the terms on the left side to create a single fraction. Identify critical points where the numerator and denominator equal zero, as these will help determine the intervals for testing the inequality. Finally, graph the solution on a real number line, marking the critical points and indicating where the inequality holds true.
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(3x - 5)/(x - 5) > 4

How does one complete this problem?
 
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We are given to solve:

$$\frac{3x-5}{x-5}>4$$

Now, one might be tempted to multiply through by $x-5$ to clear the denominator on the right side, as we would with an equation. But with inequalities, we have to be mindful of the sign of a multiplicative quantity, so our best strategy here is to subtract 4 from both sides:

$$\frac{3x-5}{x-5}-4>0$$

Now, you need to combine terms on the left, and then determine your critical numbers, which are found anywhere the numerator AND denominator is zero. Can you proceed?
 
Yes, thank you.
 

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