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Chapter 2: Problem 57

Solve each rational inequality 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 interval notation of the solution set for the given inequality \(\frac{x+4}{2 x-1} \leq 3\) comprises the valid intervals deduced from the critical values, where the inequality holds true.

Step by step solution

01

Setup Concerning the Inequality

To resolve the inequality problem given by \(\frac{x+4}{2x-1} \leq 3\), it needs to be rearranged for better computation. Thus, it becomes \(\frac{x+4}{2x-1} - 3 \leq 0\). This can be further simplified to \(\frac{x+4 - 3*(2x-1)}{2x-1} \leq 0\). After performing the algebra in the numerator by distributing and combining like terms, the inequation becomes \(\frac{-5x+7}{2x-1} \leq 0\).
02

Find the Critical Values

These values determine where the rational inequality can either be equal to 0 or undefined. For a rational inequality to equal 0, the numerator must be zero. So setting -5x + 7 = 0 and solving for x gives x = 7/5. The inequality can be undefined when the denominator equals zero. Therefore, setting 2x - 1 = 0 and solving for x gives x = 1/2.
03

Testing the Inequality in the Interval Regions

After deducing both critical values (1/2 and 7/5), the number line divides into three intervals to check: (-鈭�, 1/2), (1/2, 7/5), and (7/5, 鈭�). Choosing a number in each of these intervals to test in the inequality, and establish these intervals as part of the solution set. If the inequality is true for the test point, the interval is included in the solution set. For simplification, the inequality \(\frac{-5x+7}{2x-1}\) without the '鈮� 0' condition can be used for testing.
04

Express the Solution Set in Interval Notation

For each interval where the test point makes the inequality true, represent it in interval notation. If the interval is part of the solution set, it means the inequality holds true for all numbers in that interval. Hence, the solution in interval notation is assembled from these valid intervals.

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Most popular questions from this chapter

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x+2}-2$$

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations of \(f(x)=\frac{1}{x}\) to graph \(g.\) $$g(x)=\frac{3 x-7}{x-2}$$

Solve: \(\sqrt{x+7}-1=x\)

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-4}{x}$$

Use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$(x-2)^{2}>0$$
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