Chapter 1

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. $$ y_{1}=\frac{2 x}{x+2}, y_{2}=\frac{3}{x+4}, \text { and } y_{1}+y_{2}=1 $$

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.

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. $$\frac{2 x-1}{3}-\frac{x-5}{6}=\frac{x-3}{4}$$

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 Celsius temperature, \(C,\) to Fahrenheit temperature, \(F,\) is $$ F=\frac{9}{5} C+32 $$ 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.

Explain how to solve an equation involving absolute value.

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 2002

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.

Explain why the procedure that you explained in Exercise 120 does not apply to the equation \(|x-2|=-3\) What is the solution set for this equation?

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. $$ \begin{aligned} &y_{1}=-x^{2}+4 x-2, y_{2}=-3 x^{2}+x-1, \text { and } &y_{1}-y_{2}=0 \end{aligned} $$

In Exercises 122–133, 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 dollar per day plus 0.20 dollar per mile. Continental charges 20 dollar per day plus 0.50 dollar 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 \(10 .\)

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

In Exercises 122–133, 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 dollar with a charge of 0.08 dollar per text. Plan \(B\) has a monthly fee of 3 dollar with a charge of 0.12 dollar 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 numbers that must be excluded from the domain of each rational expression. $$ \frac{7}{2 x^{2}-8 x+5} $$

In Exercises 122–133, 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 dollar plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of 200 dollar 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.

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

In Exercises 122–133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A local bank charges 8 dollar per month plus 5 g per check. The credit union charges 2 dollar per month plus 8 g 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.

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

In Exercises 122–133, 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 dollar and it costs 0.40 dollar to produce each tape. The selling price is 2.00 dollar per tape. How many tapes must be produced and sold each week for the company to generate a profit?

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.

Solve equation by the method of your choice. $$ \frac{1}{x^{2}-3 x+2}=\frac{1}{x+2}+\frac{5}{x^{2}-4} $$

In Exercises 122–133, 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 dollar and it costs 3.00 dollar 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?

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.

Solve equation by the method of your choice. $$ \frac{x-1}{x-2}+\frac{x}{x-3}=\frac{1}{x^{2}-5 x+6} $$

In Exercises 122–133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I solve an equation that is quadratic in form, it's important to write down the substitution that I am making.

If \(x\) represents a number, write an English sentence about the number that results in an inconsistent equation.

Solve equation by the method of your choice. $$ \sqrt{2} x^{2}+3 x-2 \sqrt{2}=0 $$

In Exercises 122–133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Find \(b\) such that \(\frac{7 x+4}{b}+13=x\) has a solution set given by \(\\{-6\\}\)

Solve equation by the method of your choice. $$ \sqrt{3} x^{2}+6 x+7 \sqrt{3}=0 $$

In Exercises 122–133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least 90%. On the first four examinations, you have grades of 86%, 88%, 92%, and 84%. If the final examination counts as two grades, what must you get on the final to earn an A in the course?

Find \(b\) such that \(\frac{4 x-b}{x-5}=3\) has a solution set given by \(\varnothing\)

In a round-robin chess tournament, each player is paired with every other player once. The formula $$ N=\frac{x^{2}-x}{2} $$ models the number of chess games, \(N,\) that must be played in a round-robin tournament with \(x\) chess players. Use this formula to solve. In a round-robin chess tournament, 21 games were played. How many players were entered in the tournament?

In Exercises 122–133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. On two examinations, you have grades of 86 and 88. There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90. a. What must you get on the final to earn an \(A\) in the course? b. By taking the final, if you do poorly, you might risk the B that you have in the course based on the first two exam grades. If your final average is less than \(80,\) you will lose your \(\mathrm{B}\) in the course. Describe the grades on the final that will cause this to happen.

In Exercises \(129-132\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. To solve \(x-9 \sqrt{x}+14=0,\) we let \(\sqrt{u}=x\)

Exercises \(131-133\) will help you prepare for the material covered in the next section. Jane's salary exceeds Jim's by \(\$ 150\) per week. If \(x\) represents Jim's weekly salary, write an algebraic expression that models Jane's weekly salary.

In a round-robin chess tournament, each player is paired with every other player once. The formula $$ N=\frac{x^{2}-x}{2} $$ models the number of chess games, \(N,\) that must be played in a round-robin tournament with \(x\) chess players. Use this formula to solve. In a round-robin chess tournament, 36 games were played. How many players were entered in the tournament?