Chapter 1

Explain how to find restrictions on the variable in a rational equation.

For each planet in our solar system, its year is the time it takes the planet to revolve once around the sun. The formula $$E=0.2 x^{\frac{3}{2}}$$ models the number of Earth days in a planet's year, \(E,\) where \(x\) is the average distance of the planet from the sun, in millions of kilometers. We, of course, have 365 Earth days in our year. What is the average distance of Earth from the sun? Use a calculator and round to the nearest million kilometers.

Why should restrictions on the variable in a rational equation be listed before you begin solving the equation?

For each planet in our solar system, its year is the time it takes the planet to revolve once around the sun. The formula $$E=0.2 x^{\frac{3}{2}}$$ models the number of Earth days in a planet's year, \(E,\) where \(x\) is the average distance of the planet from the sun, in millions of kilometers. There are approximately 88 Earth days in the year of the planet Mercury. What is the average distance of Mercury from the sun? Use a calculator and round to the nearest million kilometers.

In Exercises \(115-122,\) find all values of \(x\) satisfying the given conditions. $$ y=2 x^{2}-3 x \text { and } y=2 $$

Without actually solving the equation, give a general description of how to solve \(x^{3}-5 x^{2}-x+5=0\)

In Exercises \(115-122,\) find all values of \(x\) satisfying the given conditions. $$ y=5 x^{2}+3 x \text { and } y=2 $$

In solving \(\sqrt{3 x+4}-\sqrt{2 x+4}=2,\) why is it a good idea to isolate a radical term? What if we don't do this and simply square each side? Describe what happens.

Use your graphing utility to enter each side of the equation separately under \(y_{1}\) and \(y_{2} .\) Then use the utility's TABLE or GRAPH feature to solve the equation. \(5 x+2(x-1)=3 x+10\)

What is an extraneous solution to a radical equation?

In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and 2008. Also shown is the percentage of households in which a person of faith is married to someone with no religion. (Graph cant copy) The formula $$I=\frac{1}{4} x+26$$ models the percentage of U.S. households with an interfaith marriage, \(I\), \(x\) years after 1988 . The formula $$N=\frac{1}{4} x+6$$ models the percentage of U.S. households in which a person of faith is married to someone with no religion, \(N, x\) years after 1988 Use these models to solve Exercises. a. In which years will more than 33% of U.S. households have an interfaith marriage? b. In which years will more than 14% of U.S. households have a person of faith married to someone with no religion? c. Based on your answers to parts (a) and (b), in which years will more than 33% of households have an interfaith marriage and more than 14% have a faith/no religion marriage? d. Based on your answers to parts (a) and (b), in which years will more than 33% of households have an interfaith marriage or more than 14% have a faith/no religion marriage?

Use your graphing utility to enter each side of the equation separately under \(y_{1}\) and \(y_{2} .\) Then use the utility's TABLE or GRAPH feature to solve the equation. \(2 x+3(x-4)=4 x-7\)

In Exercises \(115-122,\) find all values of \(x\) satisfying the given conditions. $$ y_{1}=x-3, y_{2}=x+8, \text { and } y_{1} y_{2}=-30 $$

Explain how to recognize an equation that is quadratic in form. Provide two original examples with your explanation.

In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and 2008. Also shown is the percentage of households in which a person of faith is married to someone with no religion. (Graph cant copy) The formula $$I=\frac{1}{4} x+26$$ models the percentage of U.S. households with an interfaith marriage, \(I\), \(x\) years after 1988 . The formula $$N=\frac{1}{4} x+6$$ models the percentage of U.S. households in which a person of faith is married to someone with no religion, \(N, x\) years after 1988 Use these models to solve Exercises. a. In which years will more than 34% of U.S. households have an interfaith marriage? b. In which years will more than 15% of U.S. households have a person of faith married to someone with no religion? c. Based on your answers to parts (a) and (b), in which years will more than 34% of households have an interfaith marriage and more than 15% have a faith/no religion marriage? d. Based on your answers to parts (a) and (b), in which years will more than 34% of households have an interfaith marriage or more than 15% have a faith/no religion marriage?

Use your graphing utility to enter each side of the equation separately under \(y_{1}\) and \(y_{2} .\) Then use the utility's TABLE or GRAPH feature to solve the equation. \(\frac{x-3}{5}-1=\frac{x-5}{4}\)

In Exercises \(115-122,\) find all values of \(x\) satisfying the given conditions. $$ y_{1}=\frac{2 x}{x+2}, y_{2}=\frac{3}{x+4}, \text { and } y_{1}+y_{2}=1 $$

Describe two methods for solving this equation: \(x-5 \sqrt{x}+4=0\)

A basic cellphone plan costs \(\$ 20\) per month for 60 calling minutes. Additional time costs \(\$ 0.40\) per minute. The formula $$C=20+0.40(x-60)$$ gives the monthly cost for this plan, \(C\), for \(x\) calling minutes, where \(x>60 .\) How many calling minutes are possible for a monthly cost of at least \(\$ 28\) and at most \(\$ 40 ?\)

Use your graphing utility to enter each side of the equation separately under \(y_{1}\) and \(y_{2} .\) Then use the utility's TABLE or GRAPH feature to solve the equation. \(\frac{2 x-1}{3}-\frac{x-5}{6}=\frac{x-3}{4}\)

In Exercises \(115-122,\) find all values of \(x\) satisfying the given conditions. $$ y_{1}=\frac{3}{x-1}, y_{2}=\frac{8}{x}, \text { and } y_{1}+y_{2}=3 $$

The formula for converting Fahrenheit temperature, \(F,\) to Celsius temperature, \(C\), is $$C=\frac{5}{9}(F-32)$$ If Celsius temperature ranges from \(15^{\circ}\) to \(35^{\circ},\) inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The model \(P=-0.18 n+2.1\) describes the number of pay phones, \(P,\) in millions, \(n\) years after \(2000,\) so I have to solve a linear equation to determine the number of pay phones in 2010.

In Exercises \(115-122,\) find all values of \(x\) satisfying the given conditions. $$ \begin{aligned} &y_{1}=2 x^{2}+5 x-4, y_{2}=-x^{2}+15 x-10, \text { and }\\\ &y_{1}-y_{2}=0 \end{aligned} $$

If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if \(h,\) the number of outcomes that result in heads, satisfies \(\left|\frac{h-50}{5}\right| \geq 1.645 .\) Describe the number of outcomes that determine an unfair coin that is tossed 100 times.

Determine whether statement makes sense or does not make sense, and explain your reasoning. Although I can solve \(3 x+\frac{1}{5}=\frac{1}{4}\) by first subtracting \(\frac{1}{5}\) from both sides, I find it easier to begin by multiplying both sides by \(20,\) the least common denominator.

In Exercises \(115-122,\) find all values of \(x\) satisfying the given conditions. $$ \begin{aligned} &y_{1}=-x^{2}+4 x-2, y_{2}=-3 x^{2}+x-1, \text { and }\\\ &y_{1}-y_{2}=0 \end{aligned} $$

Use a graphing utility and the graph's \(x\) -intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$\begin{aligned}&x^{3}+3 x^{2}-x-3=0\\\&[-6,6,1] \text { by }[-6,6,1]\end{aligned}$$

use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A truck can be rented from Basic Rental for \(\$ 50\) per day plus \(\$ 0.20\) per mile. Continental charges \(\$ 20\) per day plus \(\$ 0.50\) per mile to rent the same truck. How many miles must be driven in a day to make the rental cost for Basic Rental a better deal than Continental's?

Determine whether statement makes sense or does not make sense, and explain your reasoning. Because I know how to clear an equation of fractions, I decided to clear the equation \(0.5 x+8.3=12.4\) of decimals by multiplying both sides by 10.

In Exercises \(123-124,\) list all numbers that must be excluded from the domain of each rational expression. $$ \frac{3}{2 x^{2}+4 x-9} $$

Use a graphing utility and the graph's \(x\) -intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$\begin{aligned}&-x^{4}+4 x^{3}-4 x^{2}=0\\\&[-6,6,1] \text { by }[-9,2,1]\end{aligned}$$

use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of \(\$ 15\) with a charge of \(\$ 0.08\) per text. Plan B has a monthly fee of \(\$ 3\) with a charge of \(\$ 0.12\) per text. How many text messages in a month make plan A the better deal?

Determine whether statement makes sense or does not make sense, and explain your reasoning. Because \(x=x+5\) is an inconsistent equation, the graphs of \(y=x\) and \(y=x+5\) should not intersect.

In Exercises \(123-124,\) list all numbers that must be excluded from the domain of each rational expression. $$ \frac{7}{2 x^{2}-8 x+5} $$

Use a graphing utility and the graph's \(x\) -intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$\begin{aligned}&\sqrt{2 x+13}-x-5=0\\\&[-5,5,1] \text { by }[-5,5,1]\end{aligned}$$

use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A city commission has proposed two tax bills. The first bill requires that a homeowner pay \(\$ 1800\) plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of \(\$ 200\) plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(-7 x=x\) has no solution.

When the sum of 6 and twice a positive number is subtracted from the square of the number, 0 results. Find the number.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When checking a radical equation's proposed solution, I can substitute into the original equation or any equation that is part of the solution process.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equations \(\frac{x}{x-4}=\frac{4}{x-4}\) and \(x=4\) are equivalent.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A local bank charges \(\$ 8\) per month plus \(5 \not c\) per check. The credit union charges \(\$ 2\) per month plus \(8 \not c\) per check. How many checks should be written each month to make the credit union a better deal?

When the sum of 1 and twice a negative number is subtracted from twice the square of the number, 0 results. Find the number.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After squaring both sides of a radical equation, the only solution that I obtained was extraneous, so \(\varnothing\) must be the solution set of the original equation.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audio cassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

In Exercises \(127-130,\) solve each equation by the method of your choice. $$ \frac{1}{x^{2}-3 x+2}=\frac{1}{x+2}+\frac{5}{x^{2}-4} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The equation \(5 x^{\frac{2}{3}}+11 x^{\frac{1}{3}}+2=0 \quad\) is \(\quad\) quadratic in form, but when I reverse the variable terms and obtain \(11 x^{\frac{1}{3}}+5 x^{\frac{2}{3}}+2=0,\) the resulting equation is no longer quadratic in form.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(a\) and \(b\) are any real numbers, then \(a x+b=0\) always has one number in its solution set.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is \(\$ 3000\) and it costs \(\$ 3.00\) to produce each package of stationery. The selling price is \(\$ 5.50\) per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?

In Exercises \(127-130,\) solve each equation by the method of your choice. $$ \frac{x-1}{x-2}+\frac{x}{x-3}=\frac{1}{x^{2}-5 x+6} $$