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Chapter 2: Problem 50

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$\frac{3 x+5}{6-2 x} \geq 0$$

Short Answer

Expert verified
The solution for the rational inequality \(\frac{3 x+5}{6-2 x} \geq 0\) is presented on the number line with hollow points at -5/3 and at 3, and shaded areas representing values less than -5/3 and greater than 3. In interval notation, the solution set is \((- \infty, -\frac{5}{3}) \cup (3, \infty)\).

Step by step solution

01

Find the numerator and denominator

In the given expression \(\frac{3x + 5}{6 - 2x} \geq 0\), the numerator is \(3x + 5\) and the denominator is \(6 - 2x\).
02

Find values for which the fraction is undefined

To find the undefined points, set the denominator equal to zero and solve for \(x\). So, \(6 - 2x = 0\) which gives \(x = 3\). This is the value for which the fraction is undefined.
03

Find points where the numerator is equal to zero

Set the numerator equal to zero and solve for \(x\), so \(3x + 5 = 0\). This gives \(x = -\frac{5}{3}\). So -5/3 is a critical point.
04

Test the intervals

The intervals to be tested are \((- \infty, -\frac{5}{3}), (-\frac{5}{3}, 3), (3, + \infty)\). Choose test points in these intervals and substitute in the inequality. It is seen that the inequality holds true for the intervals \((- \infty, -\frac{5}{3})\) and \((3, \infty)\).
05

Graph and write in interval notation

The graph of the solution set is a number line with open circles at -5/3 and 3, and shading to the left of -5/3 and to the right of 3. In interval notation, the solution set is \((- \infty, -\frac{5}{3}) \cup (3, \infty)\).

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