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Chapter 3: Problem 57

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+4}{2 x-1} \leq 3 $$

Short Answer

Expert verified
The solution in interval notation for the given inequality \(\frac{x + 4}{2x - 1} \leq 3 \) is \(( -\infty, \frac{1}{2}] \cup [\frac{7}{6}, \infty )\)

Step by step solution

01

Isolate the term with variable

Subtract 3 from both sides of the inequality to isolate the rational function. After subtraction, you get \(\frac{x+ 4}{2x - 1} - 3 \leq 0 \), which can be written as \(\frac{x+4 -3(2x-1)}{2x-1} \leq 0 \). Simplification yields \(\frac{-6x +7}{2x-1} \leq 0 \).
02

Factor the expression

In this case, the expression cannot be further factored. So, move to the next step.
03

Set inequality to zero and solve

Solving the inequality \(\frac{-6x +7}{2x-1} \leq 0 \) by setting inequality to zero gives two critical points. Solving \( -6x +7 =0 \) gives \( x = \frac{7}{6} \) and solving \( 2x -1 =0 \) gives \( x = \frac{1}{2} \).
04

Test intervals

The critical points partition the number line into three intervals. Test each by choosing a test value in the interval: less than \(\frac{1}{2}\), greater than \(\frac{1}{2}\) but less than \(\frac{7}{6}\), and greater than \(\frac{7}{6}\). Substitute these values into the inequality and determine if the result is positive or negative.
05

Graph the solution on number line

To plot the graph, mark points at \( x =\frac{1}{2} \) and \( x =\frac{7}{6} \) on the number line. Then, plot the intervals for which the inequality is true (from the result of step 4). Since the inequality is \(\leq 0 \), the solutions at \( x =\frac{1}{2} \) and \( x =\frac{7}{6} \) are included in the solution set.
06

Write the solution in interval notation

Based on the graph/data from previous step, write down the intervals for which the inequality holds. Be sure to check if the points need to be included or not in the interval ( use '[ ]' if point included and '( )' if not).

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Most popular questions from this chapter

What is meant by the end behavior of a polynomial function?

Solve and graph the solution set on a number line: $$\frac{2 x-3}{4} \geq \frac{3 x}{4}+\frac{1}{2}$$ (Section 1.7, Example 5)

What are the zeros of a polynomial function and how are they found?

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=-x^{2}(x-1)(x+3)$$

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=\frac{1}{2}-\frac{1}{2} x^{4}$$
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