Chapter 2

You buy \(t\) tickets to a baseball game for a total of \(\$ 45\). Write an algebraic expression that represents the cost of each ticket.

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ \left(\frac{4 x}{3}\right)\left(\frac{3 x}{16}\right) $$

In Exercises 37-44, evaluate the algebraic expression for the given values of the variable(s). \(-3 x+2(x+y)\) (a) \(x=-2, y=2\) (b) \(x=0, y=5\)

Is there more than one way to write a verbal model? Explain.

In Exercises 45-50, an expression for the balance in an account is given. Use a guess, check, and revise strategy to determine the time (in years) necessary for the investment of \(\$ 1000\) to double. Interest rate: \(7 \%\) $$ 1000(1+0.07)^{t} $$

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ \left(12 x y^{2}\right)\left(-2 x^{3} y^{2}\right) $$

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(b^{2}-4 a b\) (a) \(a=2, b=-3\) (b) \(a=6, b=-4\)

Describe the steps that can be used to transform an equation into an equivalent equation.

In Exercises 45-50, an expression for the balance in an account is given. Use a guess, check, and revise strategy to determine the time (in years) necessary for the investment of \(\$ 1000\) to double. Interest rate: \(5 \%\) $$ 1000(1+0.05)^{t} $$

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ \left(7 r^{2} s^{3}\right)(3 r s) $$

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(a^{2}+2 a b\) (a) \(a=-2, b=3\) (b) \(a=4, b=-2\)

When dividing each side of an equation by the same quantity, why must the quantity be nonzero?

In Exercises 45-50, an expression for the balance in an account is given. Use a guess, check, and revise strategy to determine the time (in years) necessary for the investment of \(\$ 1000\) to double. Interest rate: \(6 \%\) $$ 1000(1+0.06)^{t} $$

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 2(x-2)+4 $$

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(|2 x-3 y|\) (a) \(x=2, y=3\) (b) \(x=-1, y=4\)

Describe how to solve \(x-25=250\).

In Exercises 45-50, an expression for the balance in an account is given. Use a guess, check, and revise strategy to determine the time (in years) necessary for the investment of \(\$ 1000\) to double. Interest rate: \(8 \%\) $$ 1000(1+0.08)^{t} $$

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 3(x-5)-2 $$

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(y-|-3 x+y|\) (a) \(x=-2, y=-1\) (b) \(x=7, y=3\)

In Exercises 49 and 50 , justify each step of the equation. Then identify any properties of equality used to solve the equation. $$ \begin{aligned} \frac{x}{3} &=x+1 \\ 3\left(\frac{x}{3}\right) &=3(x+1) \\ x &=3 x+3 \\ x-3 x &=3 x+3-3 x \\ x-3 x &=3 x-3 x+3 \\ -2 x &=3 \\ \frac{-2 x}{-2} &=\frac{3}{-2} \\ x &=-\frac{3}{2} \end{aligned} $$

In Exercises 45-50, an expression for the balance in an account is given. Use a guess, check, and revise strategy to determine the time (in years) necessary for the investment of \(\$ 1000\) to double. Interest rate: \(6.5 \%\) $$ 1000(1+0.065)^{t} $$

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 6(2 s-1)+s+4 $$

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(\frac{x-2 y}{x+2 y}\) (a) \(x=4, y=2\) (b) \(x=4, y=-2\)

In Exercises 49 and 50 , justify each step of the equation. Then identify any properties of equality used to solve the equation. $$ \begin{aligned} \frac{4}{5} x &=4 x-16 \\ \frac{5}{4}\left(\frac{4}{5} x\right) &=\frac{5}{4}(4 x-16) \\ x &=5 x-20 \\ x-5 x &=5 x-20-5 x \\ x-5 x &=5 x-5 x-20 \\ -4 x &=-20 \\ \frac{-4 x}{-4} &=\frac{-20}{-4} \\ x &=5 \end{aligned} $$

In Exercises 45-50, an expression for the balance in an account is given. Use a guess, check, and revise strategy to determine the time (in years) necessary for the investment of \(\$ 1000\) to double. Interest rate: \(7.5 \%\) $$ 1000(1+0.075)^{t} $$

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ (2 x-1) 2+x+9 $$

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(\frac{5 x}{y-3}\) (a) \(x=2, y=4\) (b) \(x=2, y=3\)

In Exercises 51-54, determine whether each value of \(x\) is a solution of the equation. \(\frac{2}{x}-\frac{1}{x}=1\) (a) \(x=0\) (b) \(x=\frac{1}{3}\)

The sides of a square have lengths of \(a\) centimeters. Draw the square. Draw the rectangle obtained by extending two parallel sides of the square 6 centimeters. Find expressions for the perimeter and area of each figure.

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ m-3(m-7) $$

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(\frac{-y}{x^{2}+y^{2}}\) (a) \(x=0, y=5\) (b) \(x=1, y=-3\)

In Exercises 51-54, determine whether each value of \(x\) is a solution of the equation. \(\frac{4}{x}+\frac{2}{x}=1\) (a) \(x=0\) (b) \(x=6\)

The dimensions of a rectangular lawn are 150 feet by 250 feet. The property owner buys a rectangular strip of land \(x\) feet wide along one 250 -foot side of the lawn. Draw diagrams representing the lawn before and after the purchase. Write an expression for the area of each.

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 8 l-(3 l-7) $$

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(\frac{2 x-y}{y^{2}+1}\) (a) \(x=1, y=2\) (b) \(x=1, y=3\)

In Exercises 51-54, determine whether each value of \(x\) is a solution of the equation. \(\frac{5}{x-1}+\frac{1}{x}=5\) (a) \(x=3\) (b) \(x=\frac{1}{6}\)

A bubble rises through water at a rate of about \(1.15\) feet per second. How far will the bubble rise in 5 seconds?

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ -6(2-3 x)+10(5-x) $$

\((x+2 y)(-3 x-z)\) (a) \(x=2, y=-1, z=-1\) (b) \(x=-3, y=2, z=-2\)

In Exercises 51-54, determine whether each value of \(x\) is a solution of the equation. \(\frac{3}{x-2}=x\) (a) \(x=-1\) (b) \(x=3\)

A train travels at an average speed of 50 miles per hour. How long will it take the train to travel 350 miles?

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 3(r-2 s)-5(3 r-5 s) $$

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(\frac{y z-3}{x+2 z}\) (a) \(x=0, y=-7, z=3\) (b) \(x=-2, y=-3, z=3\)

In Exercises \(55-58\), write an algebraic equation. Do not solve the equation. A student has \(n\) quarters and seven \(\$ 1\) bills totaling \(\$ 8.75\). How many quarters does the student have?

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ \frac{2}{3}(12 x+15)+16 $$

The interest a savings account earns is given by \(I=850(0.095) t\), where \(I\) is the interest the account earns after \(t\) years. Use a spreadsheet to determine the interest the account earns after 8 years. $$ \begin{array}{|c|c|r|c|} \hline \multicolumn{1}{|c|}{\text { DATA }} & \multicolumn{1}{c|}{\text { A }} & \multicolumn{1}{c|}{\text { B }} & \\ \hline & 1 & \text { Years } & \text { Interest } \\ \cline { 2 - 4 } & 2 & 1 & \\ \hline \end{array} $$

In Exercises \(55-58\), write an algebraic equation. Do not solve the equation. A school science club conducts a car wash to raise money. The club spends \(\$ 12\) on supplies and charges \(\$ 5\) per car. After the car wash, the club has a profit of \(\$ 113\). How many cars did the members of the science club wash?

Two unknown quantities in a verbal model are "Number of cherries" and "Number of strawberries." What variables would you use to represent these quantities. Explain.

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ \frac{3}{8}(4-y)-\frac{5}{2}+10 $$

The distance a car travels is given by \(d=63 t\), where \(d\) is the distance (in miles) the car travels after \(t\) hours. Use a spreadsheet to determine the distance the car travels after 7 hours. $$ \begin{array}{|r|r|c|} \hline & \multicolumn{1}{|c|}{\text { A }} & \text { B } \\ \hline 1 & \text { Hours } & \text { Distance } \\ \hline 2 & 1 & \\ \hline 3 & 2 & \\ \hline 4 & 3 & \\ \hline 5 & 4 & \\ \hline 6 & 5 & \\ \hline 7 & 6 & \\ \hline 8 & 7 & \\ \hline 9 & & \\ \hline \end{array} $$