Chapter 1

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

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(S=P+P r t\) 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. $$\frac{1}{x-4}-\frac{5}{x+2}=\frac{6}{x^{2}-2 x-8}$$

Perform the indicated operation(s) and write the result in standard form. $$ 5 \sqrt{-16}+3 \sqrt{-81} $$

write each English sentence as an equation in two variables. Then graph the equation. The \(y\) -value is three decreased by the square of 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. $$ 6(x-1)-(4-x) \geq 7 x-8 $$

Solve equation by completing the square. $$ x^{2}+4 x=12 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$2 x^{\frac{2}{3}}+7 x^{\frac{1}{3}}-15=0$$

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

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. $$\frac{6}{x+3}-\frac{5}{x-2}=\frac{-20}{x^{2}+x-6}$$

Perform the indicated operation(s) and write the result in standard form. $$ 5 \sqrt{-8}+3 \sqrt{-18} $$

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

In Exercises 51–58, solve each compound inequality. $$ 6

Solve equation by completing the square. $$ x^{2}-6 x-11=0 $$

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

Find all values of x satisfying the given conditions. $$y_{1}=5(2 x-8)-2, y_{2}=5(x-3)+3, \text { and } y_{1}=y_{2}$$

Evaluate \(x^{2}-2 x+2\) for \(x=1+i\)

In Exercises 51–58, solve each compound inequality. $$ 7

Solve equation by completing the square. $$ x^{2}-2 x-5=0 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$x^{\frac{2}{5}}+x^{\frac{1}{5}}-6=0$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(S=\frac{C}{1-r} \text { for } r\)

Find all values of x satisfying the given conditions. $$y_{1}=7(3 x-2)+5, y_{2}=6(2 x-1)+24, \text { and } y_{1}=y_{2}$$

Evaluate \(x^{2}-2 x+5\) for \(x=1-2 i\)

Graph each equation. \(y=-1 \text { (Let } x=-3,-2,-1,0,1,2, \text { and } 3 .)\)

In Exercises 51–58, solve each compound inequality. $$ -3 \leq x-2<1 $$

Solve equation by completing the square. $$ x^{2}+4 x+1=0 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$2 x-3 x^{\frac{1}{2}}+1=0$$

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

Find all values of x satisfying the given conditions. $$y_{1}=\frac{x-3}{5}, y_{2}=\frac{x-5}{4}, \text { and } y_{1}-y_{2}=1$$

Evaluate \(\frac{x^{2}+19}{2-x}\) for \(x=3 i\)

Graph each equation. \(y=\frac{1}{x}\left(\text { Let } x=-2,-1,-\frac{1}{2},-\frac{1}{3}, \frac{1}{3}, \frac{1}{2}, 1, \text { and } 2 .\right)\)

Solve equation by completing the square. $$ x^{2}+6 x-5=0 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$x+3 x^{\frac{1}{2}}-4=0$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=2 l w+2 l h+2 w h\) for \(h\)

Find all values of x satisfying the given conditions. $$y_{1}=\frac{x+1}{4}, y_{2}=\frac{x-2}{3}, \text { and } y_{1}-y_{2}=-4$$

Evaluate \(\frac{x^{2}+11}{3-x}\) for \(x=4 i\)

Graph each equation. \(y=-\frac{1}{x}\left(\text { Let } x=-2,-1,-\frac{1}{2},-\frac{1}{3}, \frac{1}{3}, \frac{1}{2}, 1, \text { and } 2 .\right)\)

In Exercises 51–58, solve each compound inequality. $$ -11<2 x-1 \leq-5 $$

Solve equation by completing the square. $$ x^{2}-5 x+6=0 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$(x-5)^{2}-4(x-5)-21=0$$

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(\frac{1}{p}+\frac{1}{q}=\frac{1}{f}\) for \(f\)

Find all values of x satisfying the given conditions. $$\begin{aligned}&y_{1}=\frac{5}{x+4}, y_{2}=\frac{3}{x+3}, y_{3}=\frac{12 x+19}{x^{2}+7 x+12}, \text { and }\\\&y_{1}+y_{2}=y_{3}\end{aligned}$$

Complex numbers are used in electronics to describe the current in an electric circuit. Ohm's law relates the current in a circuit, \(I\), in amperes, the voltage of the circuit, \(E,\) in volts, and the resistance of the circuit, \(R,\) in ohms, by the formula \(E=I R .\)Use this formula to solve Exercises \(55-56\). Find \(E,\) the voltage of a circuit, if \(I=(4-5 i)\) amperes and \(R=(3+7 i)\) ohms.

In Exercises 51–58, solve each compound inequality. $$ 3 \leq 4 x-3<19 $$

Solve equation by completing the square. $$ x^{2}+7 x-8=0 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$(x+3)^{2}+7(x+3)-18=0$$

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

Find all values of x satisfying the given conditions. $$\begin{aligned}&y_{1}=\frac{2 x-1}{x^{2}+2 x-8}, y_{2}=\frac{2}{x+4}, y_{3}=\frac{1}{x-2}, \text { and }\\\&y_{1}+y_{2}=y_{3}\end{aligned}$$

Complex numbers are used in electronics to describe the current in an electric circuit. Ohm's law relates the current in a circuit, \(I\), in amperes, the voltage of the circuit, \(E,\) in volts, and the resistance of the circuit, \(R,\) in ohms, by the formula \(E=I R .\)Use this formula to solve Exercises \(55-56\). Find \(E,\) the voltage of a circuit, if \(I=(2-3 i)\) amperes and \(R=(3+5 i)\) ohms.

In Exercises 51–58, solve each compound inequality. $$ -3 \leq \frac{2}{3} x-5<-1 $$