Chapter 1

A repair bill on a sailboat came to \(\$ 1603,\) including \(\$ 532\) for parts and the remainder for labor. If the cost of labor is \(\$ 63\) per hour, how many hours of labor did it take to repair the sailboat?

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

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.

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

Solve compound inequality. \(6< x+3<8\)

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?

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

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

Solve each equation by making an appropriate substitution. $$x^{\frac{3}{2}}-2 x^{\frac{3}{4}}+1-0$$

Solve compound inequality. \(7< x+5<11\)

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?

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

Solve each equation by making an appropriate substitution. $$x^{\frac{2}{3}}+x^{\frac{1}{3}}-6-0$$

Solve compound inequality. \(-3 \leq x-2 \leq 1\)

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

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 each equation by making an appropriate substitution. $$2 x-3 x^{\frac{1}{2}}+1-0$$

Solve compound inequality. \(-6< x-4 \leq 1\)

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?

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

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

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

Solve compound inequality. \(-11<2 x-1 \leq-5\)

In Exercises \(55-74\), solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A-I w\) for \(w\)

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

Solve each equation by making an appropriate substitution. $$(x-5)^{2}-4(x-5)-21-0$$

Solve compound inequality. \(3 \leq 4 x-3<19\)

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

Solve each equation by making an appropriate substitution. $$(x+3)^{2}+7(x+3)-18-0$$

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

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

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 \(40 .^{\prime \prime}\) 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 \(\mathrm{Rm} 15\) 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 by making an appropriate substitution. $$\left(x^{2}-x\right)^{2}-14\left(x^{2}-x\right)+24-0$$

Solve compound inequality. \(-6 \leq \frac{1}{2} x-4<-3\)

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

Solve each equation by making an appropriate substitution. $$\left(x^{2}-2 x\right)^{2}-11\left(x^{2}-2 x\right)+24-0$$

Solve absolute value inequality. \(|x|<3\)

Solve each equation in Exercises \(47-64\) by completing the square. $$2 x^{2}-7 x+3=0$$

Explain how to add complex numbers. Provide an example with your explanation.

Solve each equation by making an appropriate substitution. $$\left(y-\frac{8}{y}\right)^{2}+5\left(y-\frac{8}{y}\right)-14-0$$

Solve absolute value inequality. \(|x|<5\)

find all values of \(x\) such that \(y=0\) $$ y=\frac{1}{5 x+5}-\frac{3}{x+1}+\frac{7}{5} $$

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

Explain how to multiply complex numbers anKnow the identity \(i^{2}=-1\)d give an example.

Solve each equation by making an appropriate substitution. $$\left(y-\frac{10}{y}\right)^{2}+6\left(y-\frac{10}{y}\right)-27-0$$

Solve absolute value inequality. \(|x-1| \leq 2\)

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

What is the complex conjugate of \(2+3 i ?\) What happens when you multiply this complex number by its complex conjugate?