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Section 2.2 Expressions and Equations

Subsection Writing 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.

Example 2.12.

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 Questions Reading 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?

Subsection Evaluating Algebraic Expressions

When we evaluate an algebraic expression, we must follow the order of operations.

Example 2.13.

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 paid $72 for the tickets.

Reading Questions Reading Questions

4.

How should we write "3 times the sum of \(x\) and 12"?

Subsection Negative Numbers

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

Example 2.14.

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*}

Reading Questions Reading Questions

5.

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

6.

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

Caution 2.15.

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.

Example 2.16.

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: \(~-3-8=-3+(-8)=-11\)

Reading Questions Reading Questions

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.

Example 2.17.

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*}

Reading Questions Reading Questions

8.

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

Subsection Skills Warm-Up

Exercises Exercises

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?

Solutions Answers to Skills Warm-Up

Exercises Exercises

Exercises Homework 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.

25.
rectngle
26.
rectangles
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.

34.

Luisa's parents have agreed to pay her tuition ($800 per year) plus half her annual living expenses while she is in school. Write an expression for the amount her parents will pay if Luisa's annual expenses are \(a\) dollars.

35.

Mildred canned 80 pints of tomatoes. She kept some for herslf and divided the rest equally among her four daughters. If Mildred kept \(M\) pints, write an expression for the number of pints she gave each daughter.

36.

Moira's income is $50 more than one-third of her mother's income. Write an expression for Moira's income if her mother's income is \(I\text{.}\)

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