Chapter 1

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}+4 x=12 $$

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$6(x-1)-(4-x) \geq 7 x-8$$

By making an appropriate substitution. $$ 2 x^{\frac{2}{3}}+7 x^{\frac{1}{3}}-15=0 $$

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.

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

An HMO pamphlet contains the following recommended weight for women: "Give yourself 100 pounds for the first 5 feet plus 5 pounds for every inch over 5 feet tall." Using this description, what height corresponds to a recommended weight of 135 pounds?

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

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}-6 x-11=0 $$

Solve each compound inequality. $$6< x+3<8$$

By making an appropriate substitution. $$ x^{\frac{3}{2}}-2 x^{\frac{3}{4}}+1=0 $$

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

A job pays an annual salary of \(\$ 33,150\), which includes a holiday bonus of \(\$ 750 .\) If paychecks are issued twice a month, what is the gross amount for each paycheck?

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

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}-2 x-5=0 $$

Solve each compound inequality. $$7< x+5 <11$$

By making an appropriate substitution. $$ x^{\frac{2}{5}}+x^{\frac{1}{5}}-6=0 $$

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

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

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

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}+4 x+1=0 $$

Solve each compound inequality. $$-3 \leq x-2<1$$

By making an appropriate substitution. $$ 2 x-3 x^{\frac{1}{2}}+1=0 $$

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

The rate for a particular international person-to-person telephone call is \(\$ 0.43\) for the first minute, \(\$ 0.32\) for each additional minute, and a \(\$ 2.10\) service charge. If the cost of a call is \(\$ 5.73,\) how long did the person talk?

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

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}+6 x-5=0 $$

Solve each compound inequality. $$-6 < x-4 \leq 1$$

By making an appropriate substitution. $$x+3 x^{\frac{1}{2}}-4=0$$

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

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

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$A=lw \quad \text{for} \quad w$$

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.

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}-5 x+6=0 $$

Solve each compound inequality. $$-11<2 x-1 \leq-5$$

By making an appropriate substitution. $$(x-5)^{2}-4(x-5)-21=0$$

Find all values of \(x\) satisfying the given conditions. \(y_{1}=\frac{5}{x+4}, y_{2}=\frac{3}{x+3}, y_{3}=\frac{12 x+19}{x^{2}+7 x+12},\) and \(y_{1}+y_{2}=y_{3}\)

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$D=R T \quad \text{for} \quad R$$

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.

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}+7 x-8=0 $$

Solve each compound inequality. $$3 \leq 4 x-3<19$$

By making an appropriate substitution. $$(x+3)^{2}+7(x+3)-18=0$$

Find all values of \(x\) satisfying the given conditions. \(y_{1}=\frac{2 x-1}{x^{2}+2 x-8}, y_{2}=\frac{2}{x+4}, y_{3}=\frac{1}{x-2},\) and \(y_{1}+y_{2}=y_{3}\)

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

The mathematician Girolamo Cardano is credited with the first use (in 1545 ) of negative square roots in solving the now-famous problem, "Find two numbers whose sum is 10 and whose product is \(40 . "\) Show that the complex numbers \(5+i \sqrt{15}\) and \(5-i \sqrt{15}\) satisfy the conditions of the problem. (Cardano did not use the symbolism \(i \sqrt{15}\) or even \(\sqrt{-15 .}\) He wrote R.m 15 for \(\sqrt{-15},\) meaning "radix minus 15." He regarded the numbers \(5+\) R.m 15 and 5 - R.m 15 as "fictitious" or "ghost numbers," and considered the problem "manifestly impossible." But in a mathematically adventurous spirit, he exclaimed, "Nevertheless, we will operate." \()\)

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}+3 x-1=0 $$

Solve each compound inequality. $$-3 \leq \frac{2}{3} x-5<-1$$

By making an appropriate substitution. $$\left(x^{2}-x\right)^{2}-14\left(x^{2}-x\right)+24=0$$

Find all values of \(x\) such that \(y=0\). \(y=4[x-(3-x)]-7(x+1)\)

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

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}-3 x-5=0 $$