Chapter 1

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. $$ y=|3 x-4|+2 \text { and } y<8 $$

Solve equation by the method of your choice. $$ x^{2}-6 x+13=0 $$

If 5 times a number is decreased by \(4,\) the principal square root of this difference is 2 less than the number. Find the number(s).

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. $$ y=|2 x-5|+1 \text { and } y>9 $$

Solve equation by the method of your choice. $$ x^{2}-4 x+29=0 $$

If a number is decreased by \(3,\) the principal square root of this difference is 5 less than the number. Find the number(s).

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. \(y=7-\left|\frac{x}{2}+2\right|\) and \(y\) is at most 4

Solve equation by the method of your choice. $$ x^{2}=4 x-7 $$

Solve for \(V: r=\sqrt{\frac{3 V}{\pi h}}\)

The line graph shows the cost of inflation. What cost \(\$ 10,000\) in 1984 would cost the amount shown by the graph in subsequent years. (Graph can't copy) Here are two mathematical models for the data shown by the graph. In each formula, C represents the cost x years after 1990 of what cost \(\$ 10,000\) in 1984 $$ \begin{aligned} &\text Model \quad 1\quad C=442 x+12,969\\\ &\text Model \quad 2\quad C=2 x^{2}+390 x+13,126 \end{aligned} $$ Use these models to solve Use model 1 to determine in which year the cost will be \(\$ 26,229\) for what cost \(\$ 10,000\) in 1984

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. \(y=8-|5 x+3|\) and \(y\) is at least 6

Solve equation by the method of your choice. $$ 5 x^{2}=2 x-3 $$

Solve for \(A: r=\sqrt{\frac{A}{4 \pi}}\)

The line graph shows the cost of inflation. What cost \(\$ 10,000\) in 1984 would cost the amount shown by the graph in subsequent years. (Graph can't copy) Here are two mathematical models for the data shown by the graph. In each formula, C represents the cost x years after 1990 of what cost \(\$ 10,000\) in 1984 $$ \begin{aligned} &\text Model \quad 1\quad C=442 x+12,969\\\ &\text Model \quad 2\quad C=2 x^{2}+390 x+13,126 \end{aligned} $$ Use these models to solve Use model 1 to determine in which year the cost will be \(\$ 25,345\) for what cost \(\$ 10,000\) in 1984

In Exercises \(103-104,\) use the graph of \(y=|4-x|\) to solve each inequality. $$ |4-x|<5 $$

Solve equation by the method of your choice. $$ 2 x^{2}-7 x=0 $$

List all numbers that must be excluded from the domain of each expression. $$\frac{|x-1|-3}{|x+2|-14}$$

In Exercises \(103-104,\) use the graph of \(y=|4-x|\) to solve each inequality. $$ |4-x| \geq 5 $$

Solve equation by the method of your choice. $$ 2 x^{2}+5 x=3 $$

List all numbers that must be excluded from the domain of each expression. $$\frac{x^{3}-2 x^{2}-9 x+18}{x^{3}+3 x^{2}-x-3}$$

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

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. Use this formula to solve Exercises \(105-106\). When Michael Wilson of the Harlem Globetrotters slamdunked a basketball, his hang time for the shot was approximately 1.16 seconds. What was the vertical distance of his jump, rounded to the nearest tenth of a foot?

A company wants to increase the \(10 \%\) peroxide content of its product by adding pure peroxide \((100 \% \text { peroxide). If } x\) liters of pure peroxide are added to 500 liters of its \(10 \%\) solution, the concentration, \(C,\) of the new mixture is given by $$C=\frac{x+0.1(500)}{x+500}$$ How many liters of pure peroxide should be added to produce a new product that is \(28 \%\) peroxide?

Solve equation 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. Use this formula to solve Exercises \(105-106\). 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?

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: See Exercise \(105 .\) ) b. How many liters of pure acid should be added to produce a new mixture that is \(74 \%\) acid?

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.

Solve equation by the method of your choice. $$ \frac{2 x}{x-3}+\frac{6}{x+3}=-\frac{28}{x^{2}-9} $$

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

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.

Solve equation by the method of your choice. $$ \frac{3}{x-3}+\frac{5}{x-4}=\frac{x^{2}-20}{x^{2}-7 x+12} $$

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?

Suppose you are an algebra teacher grading the following solution on an examination: $$\begin{aligned}-3(x-6) &=2-x \\\\-3 x-18 &=2-x \\\\-2 x-18 &=2 \\\\-2 x &=-16 \\\x &=8\end{aligned}$$ You should note that \(8 \text { checks, so the solution set is } \){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.

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 Exercises \(111-112\). 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. Use the equation to solve Exercises \(111-112\). 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.

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 \(x^{3}-5 x^{2}-x+5=0\)

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

Find all values of \(x\) satisfying the given conditions. $$ y_{1}=x-1, y_{2}=x+4, \text { and } y_{1} y_{2}=14 $$

What is an extraneous solution to a radical equation?

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. $$2 x+3(x-4)=4 x-7$$

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.