Chapter 4

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt{2 x-4}+2=\sqrt{3 x+4}\) (b) \(\sqrt{2 x-4}+2>\sqrt{3 x+4}\) (c) \(\sqrt{2 x-4}+2<\sqrt{3 x+4}\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(x+3)(x-5)}{(x+1)(x-4)}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$32^{0.2}$$

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt[3]{x^{2}-2 x}=\sqrt[3]{x}\) (b) \(\sqrt[3]{x^{2}-2 x}>\sqrt[3]{x}\) (c) \(\sqrt[3]{x^{2}-2 x}<\sqrt[3]{x}\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{3 x^{2}+3 x-6}{x^{2}-x-12}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$81^{0.25}$$

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt[3]{4 x^{2}-4 x+1}=\sqrt[3]{x}\) (b) \(\sqrt[3]{4 x^{2}-4 x+1}>\sqrt[3]{x}\) (c) \(\sqrt[3]{4 x^{2}-4 x+1}<\sqrt[3]{x}\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{4 x^{2}+4 x-24}{x^{2}-3 x-10}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$\left(\frac{5}{6}\right)^{-1.3}$$

Solve each equation and inequality. (a) \(\frac{(x-2)(2)-(2 x+1)(1)}{(x-2)^{2}}=0\) (b) \(\frac{(x-2)(2)-(2 x+1)(1)}{(x-2)^{2}}<0\)

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt[4]{3 x+1}=1\) (b) \(\sqrt[4]{3 x+1}>1\) (c) \(\sqrt[4]{3 x+1}<1\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{9 x^{2}-1}{x^{2}-4}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$\left(\frac{4}{7}\right)^{-0.6}$$

Solve each equation and inequality. (a) \(\frac{\left(x^{2}-1\right)(1)-(x+1)(2 x)}{\left(x^{2}-1\right)^{2}}=0\) (b) \(\frac{\left(x^{2}-1\right)(1)-(x+1)(2 x)}{\left(x^{2}-1\right)^{2}}>0\)

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt[4]{x-15}=2\) (b) \(\sqrt[4]{x-15}>2\) (c) \(\sqrt[4]{x-15}<2\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{16 x^{2}-9}{x^{2}-9}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$\pi^{-3}$$

Solve each equation and inequality. (a) \(\frac{\left(x^{2}+1\right)(2 x)-\left(x^{2}-1\right)(2 x)}{\left(x^{2}+1\right)^{2}}=0\) (b) \(\frac{\left(x^{2}+1\right)(2 x)-\left(x^{2}-1\right)(2 x)}{\left(x^{2}+1\right)^{2}} \geq 0\)

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \((2 x-5)^{1 / 2}-2=(x-2)^{1 / 2}\) (b) \((2 x-5)^{1 / 2}-2 \geq(x-2)^{1 / 2}\) (c) \((2 x-5)^{1 / 2}-2 \leq(x-2)^{1 / 2}\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(x-3)(x+1)}{(x-1)^{2}}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$(2 \pi)^{4 / 3}$$

Solve each equation and inequality. (a) \(\frac{\left(x^{2}-1\right)(3)-(3 x-1)(2 x)}{\left(x^{2}-1\right)^{2}}=0\) (b) \(\frac{\left(x^{2}-1\right)(3)-(3 x-1)(2 x)}{\left(x^{2}-1\right)^{2}} \leq 0\)

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \((x+5)^{1 / 2}-2=(x-1)^{1 / 2}\) (b) \((x+5)^{1 / 2}-2 \geq(x-1)^{1 / 2}\) (c) \((x+5)^{1 / 2}-2 \leq(x-1)^{1 / 2}\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x(x-2)}{(x+3)^{2}}$$

Evaluate \(f(x)\) at the given \(x\). Approximate each result to the nearest hundredth. $$f(x)=x^{1.62}, \quad x=1.2$$

Solve each equation and inequality. (a) \(\frac{(2 x+1)(2 x)-\left(x^{2}+1\right)(2)}{(2 x+1)^{2}}=0\) (b) \(\frac{(2 x+1)(2 x)-\left(x^{2}+1\right)(2)}{(2 x+1)^{2}}<0\)

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\left(x^{2}+6 x\right)^{1 / 4}=2\) (b) \(\left(x^{2}+6 x\right)^{1 / 4}>2\) (c) \(\left(x^{2}+6 x\right)^{1 / 4}<2\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x}{x^{2}-9}$$

Evaluate \(f(x)\) at the given \(x\). Approximate each result to the nearest hundredth. $$f(x)=x^{-0.71}, \quad x=3.8$$

Solve each equation and inequality. (a) \(\frac{(x-1)(2 x)-\left(x^{2}\right)(1)}{(x-1)^{2}}=0\) (b) \(\frac{(x-1)(2 x)-\left(x^{2}\right)(1)}{(x-1)^{2}}>0\)

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\left(x^{2}+2 x\right)^{1 / 4}=3^{1 / 4}\) (b) \(\left(x^{2}+2 x\right)^{1 / 4}>3^{1 / 4}\) (c) \(\left(x^{2}+2 x\right)^{1 / 4}<3^{1 / 4}\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{-5}{2 x+4}$$

Evaluate \(f(x)\) at the given \(x\). Approximate each result to the nearest hundredth. $$f(x)=x^{3 / 2}-x^{1 / 2}, \quad x=50$$

In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$\frac{1}{x^{2}+1}>0$$ The numerator of the rational expression, \(1,\) is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nomegative number, \(x^{2},\) and a positive number, i. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than 0 , and this will always be true, the solution set is ( \(-\infty, \infty\) ). Use similar reasoning to solve each inequality. $$\frac{-1}{x^{2}+2}<0$$

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \((2 x-1)^{2 / 3}=x^{1 / 3}\) (b) \((2 x-1)^{2 / 3}>x^{1 / 3}\) (c) \((2 x-1)^{2 / 3}

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{-4}{3 x+9}$$

Evaluate \(f(x)\) at the given \(x\). Approximate each result to the nearest hundredth. $$f(x)=x^{5 / 4}-x^{-3 / 4}, \quad x=7$$

In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$\frac{1}{x^{2}+1}>0$$ The numerator of the rational expression, \(1,\) is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nomegative number, \(x^{2},\) and a positive number, i. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than 0 , and this will always be true, the solution set is ( \(-\infty, \infty\) ). Use similar reasoning to solve each inequality. $$\frac{5}{x^{2}+2}<0$$

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \((x-3)^{2 / 5}=(4 x)^{1 / 5}\) (b) \((x-3)^{2 / 5}>(4 x)^{1 / 5}\) (c) \((x-3)^{2 / 5}<(4 x)^{1 / 5}\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(3-x)^{2}}{(1-x)(4+x)}$$

Sketch the graph of each power function by hand, using a calculator only to evaluate \(y\)-values for your chosen \(x\) -values. On the same axes, graph \(y=x^{2}\) for comparison. In each case, \(x \geq 0\) $$f(x)=x^{1.05}$$

In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$\frac{1}{x^{2}+1}>0$$ The numerator of the rational expression, \(1,\) is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nomegative number, \(x^{2},\) and a positive number, i. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than 0 , and this will always be true, the solution set is ( \(-\infty, \infty\) ). Use similar reasoning to solve each inequality. $$\frac{-5}{x^{2}+2}>0$$

Incorporate many concepts from Chapter 3 with the method of solving equations involving roots. Work them in order. Consider the equation $$\sqrt[3]{4 x-4}=\sqrt{x+1}$$ Rewrite the equation, using rational exponents.

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(x+4)^{2}}{(x-1)(x+5)}$$

Sketch the graph of each power function by hand, using a calculator only to evaluate \(y\)-values for your chosen \(x\) -values. On the same axes, graph \(y=x^{2}\) for comparison. In each case, \(x \geq 0\) $$f(x)=x^{1.5}$$

In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$\frac{1}{x^{2}+1}>0$$ The numerator of the rational expression, \(1,\) is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nomegative number, \(x^{2},\) and a positive number, i. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than 0 , and this will always be true, the solution set is ( \(-\infty, \infty\) ). Use similar reasoning to solve each inequality. $$\frac{x^{4}+2}{-6} \leq 0$$

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(x+1)^{2}}{(x+2)(x-3)}$$

Sketch the graph of each power function by hand, using a calculator only to evaluate \(y\)-values for your chosen \(x\) -values. On the same axes, graph \(y=x^{2}\) for comparison. In each case, \(x \geq 0\) .$$f(x)=x^{2.5}$$

In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$\frac{1}{x^{2}+1}>0$$ The numerator of the rational expression, \(1,\) is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nomegative number, \(x^{2},\) and a positive number, i. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than 0 , and this will always be true, the solution set is ( \(-\infty, \infty\) ). Use similar reasoning to solve each inequality. $$\frac{x^{4}+2}{-6} \geq 0$$

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{20+6 x-2 x^{2}}{8+6 x-2 x^{2}}$$