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Chapter 2: Problem 41

Solve the inequality. Then graph the solution set. $$\frac{3 x-5}{x-5} \geq 0$$

Short Answer

Expert verified
The solution to the inequality \(\frac{3x - 5}{x - 5} \geq 0\) is \(x \in (\frac{5}{3}, 5)\)

Step by step solution

01

Identify the critical points

The critical points are the values of x which either make the numerator or the denominator of the fraction zero, or the denominator undefined. In this case, the critical points are \(x = 5\) (which makes the denominator zero and the fraction undefined) and \(x = \frac{5}{3}\) (which makes the numerator zero).
02

Test the intervals

The critical points divide the number line into intervals. Select a test point from each interval and substitute it back to the original inequality to check whether the inequality holds. Let's stick with rational numbers, for instance:- For \( x < \frac{5}{3}\), let's choose \(x = 1\). Substituting \(x = 1\) into the inequality, it simplifies to \(-2 \geq 0\), which is false.- For \(\frac{5}{3} < x < 5\), let's choose \(x = 2\). This simplifies to \(1 \geq 0\), which is true.- For \(x > 5\), let's choose \(x = 6\). The inequality simplifies to \(-1 \geq 0\), which is false.
03

Graph the solution on a number line

Sketch a number line. Make an open circle at \(x = 5\) because this value is excluded from the domain of the function. Use a solid dot at \(x = \frac{5}{3}\) because this value makes the given inequality hold as an equality. After testing the intervals, color the interval (\(\frac{5}{3}, 5)\) because they make the original inequality true.

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