Chapter 1

Solve each equation with rational exponents in Exercises \(31-40\) Check all proposed solutions. $$\left(x^{2}-3 x+3\right)^{\frac{3}{2}}-1=0$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(V=\frac{1}{3} B h\) for \(B\)

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$\frac{3}{x+4}-7=\frac{-4}{x+4}$$

Perform the indicated operations and write the result in standard form. $$ \frac{-15-\sqrt{-18}}{33} $$

In all exercises, other than \(\varnothing\), use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ \frac{x}{4}-\frac{3}{2} \leq \frac{x}{2}+1 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$x^{4}-5 x^{2}+4=0$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(I=P r t\) for \(P\)

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$\frac{8 x}{x+1}=4-\frac{8}{x+1}$$

Perform the indicated operations and write the result in standard form. $$ \sqrt{-8}(\sqrt{-3}-\sqrt{5}) $$

In all exercises, other than \(\varnothing\), use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ \frac{3 x}{10}+1 \geq \frac{1}{5}-\frac{x}{10} $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$x^{4}-13 x^{2}+36=0$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(C=2 \pi r\) for \(r\)

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$\frac{2}{x-2}=\frac{x}{x-2}-2$$

Perform the indicated operations and write the result in standard form. $$ \sqrt{-12}(\sqrt{-4}-\sqrt{2}) $$

In all exercises, other than \(\varnothing\). use interval notation to express solution sets and graph each solution set on a mumber line. In Exercises \(27-50\), solve each linear inequality. $$ 1-\frac{x}{2}>4 $$

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}-\frac{2}{3} x $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$9 x^{4}=25 x^{2}-16$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(E=m c^{2}\) for \(m\)

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$\frac{3}{2 x-2}+\frac{1}{2}=\frac{2}{x-1}$$

Perform the indicated operations and write the result in standard form. $$ (3 \sqrt{-5})(-4 \sqrt{-12}) $$

In all exercises, other than \(\varnothing\). use interval notation to express solution sets and graph each solution set on a mumber line. In Exercises \(27-50\), solve each linear inequality. $$ 7-\frac{4}{5} x<\frac{3}{5} $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$4 x^{4}=13 x^{2}-9$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(V=\pi r^{2} h\) for \(h\)

Perform the indicated operations and write the result in standard form. $$ (3 \sqrt{-7})(2 \sqrt{-8}) $$

In all exercises, other than \(\varnothing\). use interval notation to express solution sets and graph each solution set on a mumber line. In Exercises \(27-50\), solve each linear inequality. $$ \frac{x-4}{6} \geq \frac{x-2}{9}+\frac{5}{18} $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$x-13 \sqrt{x}+40=0$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(T=D+p m\) for \(p\)

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$\frac{3}{x+2}+\frac{2}{x-2}=\frac{8}{(x+2)(x-2)}$$

Perform the indicated operation(s) and write the result in standard form. $$ (2-3 i)(1-i)-(3-i)(3+i) $$

In all exercises, other than \(\varnothing,\) use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ \frac{4 x-3}{6}+2 \geq \frac{2 x-1}{12} $$

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}-\frac{1}{4} x $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$2 x-7 \sqrt{x}-30=0$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(P=C+M C\) for \(M\)

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$\frac{5}{x+2}+\frac{3}{x-2}=\frac{12}{(x+2)(x-2)}$$

Perform the indicated operation(s) and write the result in standard form. $$ (8+9 i)(2-i)-(1-i)(1+i) $$

In all exercises, other than \(\varnothing,\) use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ 4(3 x-2)-3 x<3(1+3 x)-7 $$

Solve equation by completing the square. $$ x^{2}+6 x=7 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$x^{-2}-x^{-1}-20=0$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=\frac{1}{2} h(a+b)\) for \(a\)

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$\frac{2}{x+1}-\frac{1}{x-1}=\frac{2 x}{x^{2}-1}$$

Perform the indicated operation(s) and write the result in standard form. $$ (2+i)^{2}-(3-i)^{2} $$

Write each English sentence as an equation in two variables. Then graph the equation. The \(y\) -value is four more than twice the \(x\) -value.

In all exercises, other than \(\varnothing,\) use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ 3(x-8)-2(10-x)>5(x-1) $$

Solve equation by completing the square. $$ x^{2}+6 x=-8 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$x^{-2}-x^{-1}-6=0$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=\frac{1}{2} h(a+b)\) for \(b\)

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$\frac{4}{x+5}+\frac{2}{x-5}=\frac{32}{x^{2}-25}$$

Perform the indicated operation(s) and write the result in standard form. $$ (4-i)^{2}-(1+2 i)^{2} $$

write each English sentence as an equation in two variables. Then graph the equation. The \(y\) -value is the difference between four and twice the \(x\) -value.

In all exercises, other than \(\varnothing,\) use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ 5(x-2)-3(x+4) \geq 2 x-20 $$