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

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SUMMARY

The inequality \(\frac{3x-5}{x-5} > 4\) can be solved by first rearranging it to \(\frac{3x-5}{x-5} - 4 > 0\). This requires combining terms on the left side and identifying critical points where the numerator and denominator equal zero. The critical numbers are essential for determining the intervals to test for the solution. The solution involves analyzing these intervals to graph the inequality on a real number line.

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  • Understanding of rational inequalities
  • Knowledge of critical points in algebra
  • Ability to manipulate algebraic expressions
  • Familiarity with graphing on a number line
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  • Learn how to find critical points in algebraic expressions
  • Research methods for graphing inequalities on a number line
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Students in algebra courses, educators teaching inequality concepts, and anyone looking to improve their skills in solving and graphing rational inequalities.

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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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