Chapter 1

Use the quadratic formula and a calculator to find all real solutions, correct to three decimals. \(x^{2}-1.800 x+0.810=0\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 5 x^{2}-125=0 $$

Thickness of a Laminate A company manufactures industrial laminates (thin nylon-based sheets) of thickness 0.020 in, with a tolerance of 0.003 in. (a) Find an inequality involving absolute values that describes the range of possible thickness for the laminate. (b) Solve the inequality you found in part (a).

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{4 x}{2 x+3}>2 $$

Find all solutions of the equation and express them in the form \(a+b i .\) $$ x^{2}+9=0 $$

1–54 ? Find all real solutions of the equation. $$ \sqrt{x+\sqrt{x+2}}=2 $$

Use the quadratic formula and a calculator to find all real solutions, correct to three decimals. \(2.232 x^{2}-4.112 x=6.219\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ x^{2}+16=0 $$

Range of Height The average height of adult males is 68.2 in, and 95\(\%\) of adult males have height \(h\) that satisfies the inequality $$ \left|\frac{h-68.2}{2.9}\right| \leq 2 $$ Solve the inequality to find the range of heights.

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ -2<\frac{x+1}{x-3} $$

Dimensions of a Track A running track has the shape shown in the figure, with straight sides and semicircular ends. If the length of the track is 440 yd and the two straight parts are each 110 yd long, what is the radius of the semicircular parts (to the nearest yard)?

Find all solutions of the equation and express them in the form \(a+b i .\) $$ 9 x^{2}+4=0 $$

1–54 ? Find all real solutions of the equation. $$ \sqrt{1+\sqrt{x+\sqrt{2 x+1}}}=\sqrt{5+\sqrt{x}} $$

Use the quadratic formula and a calculator to find all real solutions, correct to three decimals. \(12.714 x^{2}+7.103 x=0.987\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 6 x^{2}+100=0 $$

Using Distances to Solve Absolute Value Inequalities Recall that \(|a-b|\) is the distance between \(a\) and \(b\) on the number line. For any number \(x\) , what do \(|x-1|\) and \(|x-3|\) represent? Use this interpretation to solve the inequality \(|x-1|<|x-3|\) geometrically. In general, if \(a

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{2 x+1}{x-5} \leq 3 $$

Find all solutions of the equation and express them in the form \(a+b i .\) $$ x^{2}-4 x+5=0 $$

55–64 ? Find all solutions, real and complex, of the equation. $$ x^{3}=1 $$

Solve the equation for the indicated variable. \(h=\frac{1}{2} g t^{2}+v_{0} t ; \quad\) for \(t\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ (x+2)^{2}=4 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{3+x}{3-x} \geq 1 $$

Find all solutions of the equation and express them in the form \(a+b i .\) $$ x^{2}+2 x+2=0 $$

55–64 ? Find all solutions, real and complex, of the equation. $$ x^{4}-16=0 $$

Solve the equation for the indicated variable. \(S=\frac{n(n+1)}{a} ;\) for \(n\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 3(x-5)^{2}=15 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{4}{x}

Historical Research Read the biographical notes on Pythagoras (page \(292 ),\) Euclid (page 75\()\) , and Archimedes (page 796\()\) . Choose one of these mathematicians and find out more about him from the library or on the Internet. Write a short essay on your findings. Include both biographical information and a description of the mathematics for which he is famous.

Find all solutions of the equation and express them in the form \(a+b i .\) $$ x^{2}+x+1=0 $$

55–64 ? Find all solutions, real and complex, of the equation. $$ x^{3}+x^{2}+x=0 $$

Solve the equation for the indicated variable. \(A=2 x^{2}+4 x h ; \quad\) for \(x\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ x^{3}=27 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{x}{x+1}>3 x $$

Find all solutions of the equation and express them in the form \(a+b i .\) $$ x^{2}-3 x+3=0 $$

55–64 ? Find all solutions, real and complex, of the equation. $$ x^{4}+x^{3}+x^{2}+x=0 $$

Solve the equation for the indicated variable. \(A=2 \pi r^{2}+2 \pi r h ; \quad\) for \(r\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ x^{5}+32=0 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ 1+\frac{2}{x+1} \leq \frac{2}{x} $$

Find all solutions of the equation and express them in the form \(a+b i .\) $$ 2 x^{2}-2 x+1=0 $$

55–64 ? Find all solutions, real and complex, of the equation. $$ x^{4}-6 x^{2}+8=0 $$

Solve the equation for the indicated variable. \(\frac{1}{s+a}+\frac{1}{s+b}=\frac{1}{c} ; \quad\) for \(s\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ x^{4}-16=0 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{3}{x-1}-\frac{4}{x} \geq 1 $$

Find all solutions of the equation and express them in the form \(a+b i .\) $$ 2 x^{2}+3=2 x $$

55–64 ? Find all solutions, real and complex, of the equation. $$ x^{3}+3 x^{2}+9 x+27=0 $$

Solve the equation for the indicated variable. \(\frac{1}{r}+\frac{2}{1-r}=\frac{4}{r^{2}} ; \quad\) for \(r\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 64 x^{6}=27 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{6}{x-1}-\frac{6}{x} \geq 1 $$

Find all solutions of the equation and express them in the form \(a+b i .\) $$ t+3+\frac{3}{t}=0 $$

55–64 ? Find all solutions, real and complex, of the equation. $$ x^{6}-9 x^{3}+8=0 $$