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Chapter 2: Problem 49

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$\frac{4-2 x}{3 x+4} \leq 0$$

Short Answer

Expert verified
The solution set in interval notation is \((-\infty, -4/3] \cup (2, \infty)\), and the graph consists of a line from \(-\infty\) to \(-4/3\) (with a closed circle on \(-4/3\)) and from \(2\) to \(\infty\) (with an open circle on \(2\))

Step by step solution

01

Find the undefined values of x

To find the values of \(x\) that make this expression undefined, we set the denominator equals to zero and solve the equation for \(x\). Hence solve for: \(3x+4 = 0\).
02

Solve for x

Subtract 4 from both sides to get \(3x = -4\), then divide each side by 3 to solve for \(x\). The solution is \(x = -4/3\). This point divides the real number line into two intervals, which are \(-\infty,-4/3) and (-4/3,\infty\).
03

Find the zero of the rational expression

To find when the whole expression is equal to 0, set the numerator equals to zero and solve for \(x\). Hence solve: \(4 - 2x = 0\).
04

Solve for x to find the zero

Subtract 4 from both sides to get \(-2x = -4\), then divide each side by -2 to solve for \(x\). The solution is \(x = 2\). Now we have another point which further divides the real number line into the following intervals: \(-\infty, -4/3), (-4/3,2), (2, \infty)\)
05

Analyze for sign change in each interval

Select any number within each interval. For example, select the number -2 from the first interval, the number 0 from the second interval, and the number 3 from the third interval, and substitute these numbers into the original inequality. If the inequality is satisfied, then every point in that interval is part of the solution set. If the inequality is not satisfied, that interval is not included.
06

Express the solution in interval notation and draw the number line

Draw the number line and make sure to indicate the undefined and zeros points. If the inequality includes zero, it means the interval is closed (included in the solution set), to be indicated with a filled circle. If it is not included, use an open circle. Write the open and closed intervals as the solution in interval notation

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