## Section2.2Expressions and Equations

### SubsectionWriting Algebraic Expressions

When we write an algebraic expression, we use the same operations on a variable that we would use to calculate with a specific number. Writing down the expression for a specific numerical value can help us write an algebraic expression.

Alida keeps $100 in cash from her weekly paycheck, and deposits 40% of the remainder in her savings account. If Alida's paycheck is $p\text{,}$ write an expression for the amount she deposits in savings. Solution How would we calculate Alida's deposit if we knew her paycheck? Suppose Alida's paycheck is$500. First she subtracts $100 from that amount to get $500-100\text{,}$ and then she takes 40% of the remainder for savings: \begin{equation*} 0.40(\alert{500}-100)~~~~~\blert{\text{Perform operations inside parentheses first.}} \end{equation*} If her paycheck is $p$ dollars, we perform the same operations on $p$ instead of on 500. The expression is thus \begin{equation*} 0.40(\alert{p}-100) \end{equation*} #### Reading QuestionsReading Questions ###### 1. We write an expression with variables using the same operations we would use to calculate with a . ###### 2. In Example 2.12, how does Alida calculate the amount to deposit in savings? How do we write that in an algebraic expression? ### SubsectionEvaluating Algebraic Expressions When we evaluate an algebraic expression, we must follow the order of operations. ###### Example2.13. Four students bought concert tickets with their$20 student-discount coupons.

1. If the regular price of a ticket is $t$ dollars, write an expression for the total amount the four students paid.
2. How much did the students pay if the regular price of a ticket is 38? Solution 1. The discount price of one ticket is $t-20$ dollars. So four tickets cost \begin{equation*} 4(t-20)~~\text{dollars} \end{equation*} 2. We evaluate the expression in part (a) for $t=\alert{38}\text{.}$ \begin{equation*} \begin{aligned} 4(t-20) \amp = 4(\alert{38}-20) \amp \blert{\text{Simplify inside parentheses first.}}\\ \amp = 4(18)=72 \end{aligned} \end{equation*} The students paid72 for the tickets.

###### 3.

Explain why we need parentheses in Example 2.13.

###### 4.

How should we write "3 times the sum of $x$ and 12"?

### SubsectionNegative Numbers

The order of operations applies to signed numbers. Operations inside parentheses or other grouping devices should be performed first.

###### Example2.14.

Simplify $~~4-3-[-6+(-5)-(-2)]$

Solution

Perform the operations inside brackets first. Simplify each step by rewriting subtractions as equivalent additions.

\begin{align*} \amp 4-3- [-6+\blert{(-5)-(-2)}] \amp\amp \blert{-5-(-2)=-5+2}\\ =\,\amp 4-3-[\blert{-6-5+2}] \amp\amp \blert{-6-5=-11;~-11+2=-9}\\ =\,\amp 4 \blert{-3-[-9]} \amp\amp \blert{\text{Rewrite as an addition.}}\\ =\,\amp 4-3+9 = 10 \amp\amp \blert{\text{Add from left to right.}} \end{align*}

###### 5.

Which operation is performed first in the expression $6-4x\text{?}$

###### 6.

What is the first step in evaluating the expression $2(18-x)\text{?}$

###### Caution2.15.

When using negative numbers, we must be careful to distinguish between products and sums. In Example 2.16, note how the parentheses and minus signs are used in each expression.

###### Example2.16.

Simplify each expression.

1. $3(-8)$
2. $3-(-8)$
3. $3-8$
4. $-3-8$
Solution
1. This expression is a product: $~3(-8)=-24$

2. This is a subtraction. We follow the rule for subtraction by changing the sign of the second number and then adding: $~3-(-8)=3+8=11$

3. This is an addition; the negative sign in front of tells us that we are adding $-8$ to $3\text{.}$ Thus, $~3-8=3+(-8)=-5$

4. This is also an addition: $~-3-8=-3+(-8)=-11$

###### 7.

What is wrong with this calculation: $4-3+9=4-12=-8\text{?}$

###### Look Closer.

When we evaluate an algebraic expression at a negative number, we enclose the negative numbers in parentheses. This will help prevent us from confusing multiplication with subtraction.

###### Example2.17.

Evaluate $2x-3xy$ for $x=-5$ and $y=-2\text{.}$

Solution

We substitute $\alert{-5}$ for $x$ and $\blert{-2}$ for $y\text{,}$ then follow the order of operations.

\begin{align*} 2x-3xy \amp = 2(\alert{-5})-3(\alert{-5})(\blert{-2}) \amp \blert{\text{ Do multiplications first.}}\\ \amp = -10-(-15)(-2)\\ \amp = -10-30=-40 \end{align*}

###### 8.

When we evaluate an expression at a negative number, we should the negative number in .

### SubsectionSkills Warm-Up

#### ExercisesExercises

Choose the correct algebraic expression for each of the following situations.

\begin{equation*} \begin{aligned} \amp n+12 \amp \amp 12n \amp \amp \dfrac{n}{12}\\ \amp \dfrac{12}{n} \amp \amp 12-n \amp \amp n-12 \end{aligned} \end{equation*}
###### 1.

Helen bought $n$ packages of tulip bulbs. If each package contains 12 bulbs, how many bulbs did she buy?

###### 2.

Henry bought a package of $n$ gladiolus bulbs, then bought 12 loose bulbs. How many bulbs did he buy?

###### 3.

Together Karen and Dave sold 12 tickets to the spring concert. If Karen sold $n$ tickets, how many did Dave sell?

###### 4.

Together Karl and Diana collected $n$ used books for the book sale. If Karl collected 12 books, how many did Diana collect?

###### 5.

Greta made $n$ dollars last week. If she worked for 12 hours, how much did she make per hour?

###### 6.

Gert jogged for 12 minutes. If she jogged $n$ miles, how many minutes does it take her to jog 1 mile?

### ExercisesHomework 2.2

###### 1.

The perimeter of a rectangle of length $l$ and width $w$ is given by

\begin{equation*} P=2l+2w \end{equation*}

Find the perimeter of a rectangular meeting hall with dimensions 8.5 meters by 6.4 meters.

###### 2.

The area of a trapezoid with bases $B$ and $b$ and height $h$ is given by

\begin{equation*} A=\dfrac{1}{2}(B+b)h \end{equation*}

Find the area of a trapezoid whose bases are 9 centimeters and 7 centimeters and whose height is 3 centimeters.

###### 3.

If a company sells $n$ items at a cost of $c$ dollars each and sells them at a price of $p$ dollars each, the company's profit is given by

\begin{equation*} P=n(p-c) \end{equation*}

Find the profit earned by a manufacturer of bicycle equipment by selling 300 bicycle helmets that cost $32 each to produce and sell for$50 apiece.

###### 4.

The temperature in degrees Celsius ($\degree$C) is given by

\begin{equation*} C=\dfrac{5}{9}(F-32)\text{,} \end{equation*}

where $F$ stands for the temperature in degrees Fahrenheit ($\degree$F). Find normal body temperature in degrees Celsius if normal temperature is $98.6 \degree$ Fahrenheit.

For Problems 5–8, simplify by following the order of operations.

###### 5.
$-18-[8-12-(-4)]$
###### 6.
$3-(-6+2)+(-1-4)$
###### 7.
$-7+[-8-(-2)]-[6+(-4)]$
###### 8.
$0-[5-(-1)]+[-6-3]$
###### 9.

Find each product.

1. $(-2)(3)(4)$
2. $(-2)(-3)(4)$
3. $(-2)(-3)(-4)$
4. $(-2)(-3)(4)(2)$
5. $(-2)(-3)(-4)(2)$
6. $(-2)(-3)(-4)(-2)$
###### 10.

Use your results from Problem 9 to complete the statements:

1. The product of an odd number of negative numbers is .
2. The product of an even number of negative numbers is .

For Problems 11–14, use the order of operations to simplify the expression.

###### 11.
1. $-2(-3)-4$
2. $5(-4)-3(-6)$
###### 12.
1. $(-4-3)(-4+3)$
2. $-3(8)-6(-2)-5(2)$
###### 13.
1. $\dfrac{15}{-3}-\dfrac{4-8}{8-12}$
2. $\dfrac{4-2(-5)}{-4+3(-1)}$
###### 14.
1. $12.6-0.32(0.25)(4.2-8.7)$
2. $(5.8-2.6)(-2.5)(-0.6+3)$
###### 15.

Simplify each expression.

1. $-3(-4)(-5)$
2. $-3(-4))-5$
3. $-3(-4-5)$
4. $-3-(-4-5)$
5. $-3-(-4)(-5)$
6. $(-3-4)(-5)$
###### 16.

Simplify each expression.

1. $24 \div 6-2$
2. $24 \div (-6-2)$
3. $24 \div (-6)-2$
4. $24-6 \div (-2)$
5. $24 \div (-6) \div (-2)$
6. $24 \div (-6 \div 2)$

For Problems 17–22, evaluate for the given values.

###### 17.

$15-x-y~~~~$ for $x=-6,~y=8$

###### 18.

$p-(4-m)~~~~$ for $p=-2,~m=-6$

###### 19.

$12x-3xy~~~~$ for $x=-3,~y=2$

###### 20.

$\dfrac{y-3}{x-4}~~~~$ for $x=-9,~y=2$

###### 21.

$\dfrac{1}{2}t(t-1)~~~~$ for $t=\dfrac{2}{3}$

###### 22.

$\dfrac{x-m}{s}~~~~$ for $x=4,~m=\dfrac{9}{4},~s=\dfrac{3}{2}$

###### 23.
1. Evaluate each expression for $x=-5\text{.}$ What do you notice?

1. $\dfrac{-3}{4}x$
2. $\dfrac{-3x}{4}$
3. $-0.75x$
2. Does $\dfrac{-8}{5}x=\dfrac{-8}{5x}\text{?}$ Support your answer with examples.
###### 24.

Which of the following are equivalent to $-\dfrac{4}{9}x\text{?}$

1. $\dfrac{4}{9}(-x)$
2. $\dfrac{-4x}{-9}$
3. $\dfrac{-4}{9x}$
4. $\dfrac{-4x}{9}$
5. $\dfrac{-4}{9}(-x)$

For Problems 25–26, write an algebraic expression for area or perimeter.

###### 27.

Salewa saved $5000 to live on while going to school full time. She spends$200 per week on living expenses.

1. How much of Salewa's savings will be left after 3 weeks?
2. Describe in words how you calculated your answer to part (a).
3. Fill in the table below.
 Number of weeks $2$ $4$ $5$ Calculation $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ Savings left $\hphantom{00}$ $\hphantom{00}$ $\hphantom{00}$
4. Write an equation that gives Salewa's savings, $S\text{,}$ after $w$ weeks.
###### 28.

To calculate how much state income tax she owes, Francine subtracts $2000 from her income, and then takes 12% of the result. 1. What is Francine's state income tax if her income is$8000?
2. Describe in words how you calculated your answer to part (a).
3. Fill in the table below.
 Income $5000$ $7000$ $12,000$ Calculation $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ State tax $\hphantom{00}$ $\hphantom{00}$ $\hphantom{00}$
4. Write an equation that gives Francine's state income tax $T\text{,}$ if her income is $I\text{.}$

For Problems 29–32,

1. Choose a variable for the unknown quantity and write an algebraic expression.
2. Evaluate the expression for the given values.
###### 29.
1. Three inches less than twice the width
2. The width is 13 inches.
###### 30.
1. Twenty dollars more than 40% of the principal
2. The principal is $500. ###### 31. 1.$8 times the number of children's tickets subtracted from 150
2. There are 83 children's tickets
###### 32.
1. One-third of $50 less than the profit 2. The profit is$500

For Problems 33–40, write an algebraic expression.

###### 33.

The oven temperature started at $65 \degree$ and is rising at $30 \degree$ per minute. Write an expression for the oven temperature after $t$ minutes.

###### 37.

Otis buys 200 pounds of dog food and uses 15 pounds a week for his dog Ralph. Write an expression for the amount of dog food Otis has left after $w$ weeks.

###### 38.

Renee receives $600 for appearing in a cola commercial, plus a residual of$80 each time the commercial is aired. Write an expression for Renee's earnings if the commercial plays $t$ times.

###### 39.

Digby bought scuba diving gear by making a \$50 down payment and arranging to pay the balance in ten equal installments. If the total cost of the gear is $C$ dollars, write an expression for the amount of each installment.

###### 40.

Each passengers and three crew members on a small airplane is allowed 35 pounds of luggage. Write an expression for the weight of the luggage if there are $x$ passengers.

For Problems 41–42,

1. Write an algebraic expression.
2. Make a table for the expression showing at least two positive values for the variable and two negative values.
###### 41.

The ratio of a number to 4 more than the number

###### 42.

The sum of and 5, times the difference of and 5