Chapter 1

(graph can't copy) Suppose that \(x\) liters of pure acid are added to 200 liters of a \(35 \%\) acid solution. a. Write a formula that gives the concentration, \(C,\) of the new mixture. (Hint: Sce Exercise 105 .) b. How many liters of pure acid should be added to produce a new mixture that is \(74 \%\) acid?

Solve each equation in Exercises \(83-108\) by the method of your choice. $$\frac{1}{x}+\frac{1}{x+3}=\frac{1}{4}$$

A basketball player's hang time is the time spent in the air when shooting a basket. The formula \(t-\frac{\sqrt{d}}{2}\) models hang time, \(t\), in seconds, in terms of the vertical distance of \(a\) player's jump, \(d,\) in feet. If hang time for a shot by a professional basketball player is 0.85 second, what is the vertical distance of the jump, rounded to the nearest tenth of a foot?

When 3 times a number is subtracted from \(4,\) the absolute value of the difference is at least \(5 .\) Use interval notation to express the set of all numbers that satisfy this condition.

What is a linear equation in one variable? Give an example of this type of equation.

Solve each equation in Exercises \(83-108\) by the method of your choice. $$\frac{2 x}{x-3}+\frac{6}{x+3}=-\frac{28}{x^{2}-9}$$

When 4 times a number is subtracted from \(5,\) the absolute value of the difference is at most \(13 .\) Use interval notation to express the set of all numbers that satisfy this condition.

Suppose that you solve \(\frac{x}{5}-\frac{x}{2}-1\) by multiplying both sides by 20 rather than the least common denominator (namely, 10 ). Describe what happens. If you get the correct solution, why do you think we clear the equation of fractions by multiplying by the least common denominator?

Solve each equation in Exercises \(83-108\) by the method of your choice. $$\frac{3}{x-3}+\frac{5}{x-4}=\frac{x^{2}-20}{x^{2}-7 x+12}$$

Suppose you are an algebra teacher grading the following solution on an examination: $$ \begin{array}{c} -3(x-6)=2-x \\ -3 x-18=2-x \\ -2 x-18=2 \\ -2 x=-16 \\ x=8 \end{array} $$ You should note that 8 checks, so the solution set is \(|8|\). The student who worked the problem therefore wants full credit. Can you find any errors in the solution? If full credit is 10 points, how many points should you give the student? Justify your position.

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

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 Use the equation to solve. 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.

What is an identity? Give an example.

What is a conditional equation? Give an example.

What is an inconsistent equation? Give an example.

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

Find all values of \(x\) satisfying the given conditions. $$y=2 x^{2}-3 x \text { and } y=2$$

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. $$ 5 x+2(x-1)=3 x+10 $$

Find all values of \(x\) satisfying the given conditions. $$y=5 x^{2}+3 x \text { and } y=2$$

The formula $$1-\frac{1}{4^{x}+26}$$ 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 \(\overline{l 9} 88\). Use these models to solve. 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 houscholds 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?

The formula $$1-\frac{1}{4^{x}+26}$$ 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 \(\overline{l 9} 88\). Use these models to solve. 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?

Explain how to recognize an equation that is quadratic in form. Provide two original examples with your explanation. Describe two methods for solving this equation: \(x-5 \sqrt{x}+4-0\)

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. $$ \frac{x-3}{5}-1-\frac{x-5}{4} $$

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

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 \(\$ 40 ?\)

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

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

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

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

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 each 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.

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

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?

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

determine whether each 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

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 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 \(\mathrm{B}\) 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?

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

determine whether each 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.

List all mumbers that must be excluded from the domain of each rational expression. $$\frac{3}{2 x^{2}+4 x-9}$$

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?

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

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.

List all mumbers that must be excluded from the domain of each rational expression. $$\frac{7}{2 x^{2}-8 x+5}$$

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 e per check. The credit union charges \(\$ 2\) per month plus 8 d per check. How many checks should be written each month to make the credit union a better deal?

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.