8.1: Use the Language of Algebra

Chapter 8: Learning Outcomes

By the end of this chapter, you should be able to:

  • Explain the use of variables.
  • Evaluate algebraic expressions using substitution.
  • Combine like terms and remove parentheses.
  • Solve first-degree equations in one variable.

Use Variables & Algebraic Symbols

Greg and Alex have the same birthday but were born in different years. This year, Greg is 20 years old, and Alex is 23, so Alex is 3 years older than Greg. When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are variable, and the three is a constant. The ages change or vary, so age is a variable. The 3 years between them always stay the same, so the age difference is the constant.

In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age g. Then, we could use [latex]g+3[/latex] to represent Alex’s age (see Table 8.1.1).

Table 8.1.1: Greg’s Age vs. Alex’s Age
Greg’s Age Alex’s Age
12 15
20 23
35 38
g [latex]g+3[/latex]

Letters are used to represent variables. Letters often used for variables are x, y, a, b, and c.

Variables and constants:

A variable is a letter that represents a number or quantity whose value may change.

A constant is a number whose value always stays the same.

To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. Table 8.1.2 summarizes them, along with the words we use for the operations and the result.

Table 8.1.2: Algebraic Symbols
Operation Notation Say The Result Is
Addition [latex]a+b[/latex] a plus b The sum of a and b
Subtraction [latex]a-b[/latex] a minus b The difference of a and b
Multiplication [latex]a\cdot b, (a)(b), (a)b, a(b)[/latex] a times b The product of a and b
Division [latex]a\div b, a/b, \dfrac{a}{b}, b\enclose{longdiv}{a}[/latex] a divided by b The quotient of a and b

In algebra, the cross symbol, ×, is not used to show multiplication because that symbol may cause confusion. Does 3xy mean [latex]3 \times y[/latex] (three times y) or [latex]3 \cdot x \cdot y[/latex] (three times x times y)? To make it clear, use • or parentheses for multiplication.

We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.

  • The sum of 5 and 3 means add 5 plus 3, which we write as [latex]5+3[/latex].
  • The difference of 9 and 2 means subtract 9 minus 2, which we write as [latex]9-2[/latex].
  • The product of 4 and 8 means multiply 4 times 8, which we can write as [latex]4\cdot 8[/latex].
  • The quotient of 20 and 5 means divide 20 by 5, which we can write as [latex]20 \div 5[/latex].

Example A

Translate from algebra to words:

  1. [latex]12+14[/latex]
  2. [latex](30)(5)[/latex]
  3. [latex]64 \div 8[/latex]
  4. [latex]x-y[/latex]

 

a. [latex]\boldsymbol{12+14}[/latex]

12 plus 14.
The sum of twelve and fourteen.

 

b. [latex]\boldsymbol{(30)(5)}[/latex]

30 times 5.
The product of thirty and five.

 

c. [latex]\boldsymbol{64 \div 8}[/latex]

64 divided by 8.
The quotient of sixty-four and eight.

 

d. [latex]\boldsymbol{x-y}[/latex]

x minus y.
The difference of x and y.

Exercise 1

Translate from algebra to words.

  1. [latex]18+11[/latex]
  2. [latex](27)(9)[/latex]
  3. [latex]84\div 7[/latex]
  4. [latex]p-q[/latex]

Exercise 1 Answers

  1. 18 plus 11
    The sum of eighteen and eleven
  2. 27 times 9
    The product of twenty-seven and nine
  3. 84 divided by 7;
    The quotient of eighty-four and seven
  4. p minus q
    The difference of p and q

Exercise 2

Translate from algebra to words.

  1. [latex]47-19[/latex]
  2. [latex]72\div 9[/latex]
  3. [latex]m+n[/latex]
  4. [latex](13)(7)[/latex]

Exercise 2 Answers

  1. 47 minus 19
    The difference of forty-seven and nineteen
  2. 72 divided by 9
    The quotient of seventy-two and nine
  3. m plus n
    The sum of m and n
  4. 13 times 7
    The product of thirteen and seven

When two quantities have the same value, we say they are equal and connect them with an equal sign.

Equality symbol:

[latex]a = b[/latex] is read as a is equal to b.

The symbol = is called the equal sign.

An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities.

Remember: On the number line, the numbers get larger as they go from left to right.

So, if we know that b is greater than a, it means that b is to the right of a on the number line. We use the symbols < and > for inequalities.

Inequality:

[latex]a < b[/latex] is read a is less than b

a is to the left of b on the number line (see Figure 8.1.1).

 

The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.
Figure 8.1.1

[latex]a > b[/latex] is read a is greater than b

a is to the right of b on the number line (see Figure 8.1.2).

The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.
Figure 8.1.2

The expressions a < b and a > b can be read from left-to-right or right-to-left, though, in English, we usually read from left-to-right. In general:

  • a < b is equivalent to b > a.
    • E.g. 7 < 11 is equivalent to 11 > 7.
  • a > b is equivalent to b < a.
    • E.g. 17 > 4 is equivalent to 4 < 17.

When we write an inequality symbol with a line under it, such as [latex]a\le b[/latex], it means [latex]a < b \text{ or } a = b[/latex]. We read this as a is less than or equal to b. Also, if we put a slash through an equal sign, [latex]\ne[/latex], it means not equal.

We summarize the symbols of equality and inequality below in Table 8.1.3.

Table 8.1.3: Symbols of Equality and Inequality
Algebraic Notation Say
[latex]a=b[/latex] a is equal to b
[latex]a\ne b[/latex] a is not equal to b
[latex]a < b[/latex] a is less than b
[latex]a > b[/latex] a is greater than b
[latex]a\le b[/latex] a is less than or equal to b
[latex]a\ge b[/latex] a is greater than or equal to b

Symbols < and >:

The symbols < and > each have a smaller side and a larger side.

  • smaller side < larger side
  • larger side > smaller side

The smaller side of the symbol faces the smaller number, and the larger one faces the larger number.

Example B

Translate from algebra to words:

  1. [latex]20\le 35[/latex]
  2. [latex]11\ne 15-3[/latex]
  3. [latex]9 > 10\div 2[/latex]
  4. [latex]x+2 < 10[/latex]

 

a. [latex]\boldsymbol{20\le 35}[/latex]

20 is less than or equal to 35.

 

b. [latex]\boldsymbol{11\ne 15-3}[/latex]

11 is not equal to 15 minus 3.

 

c. [latex]\boldsymbol{9 > 10\div 2}[/latex]

9 is greater than 10 divided by 2.

 

d. [latex]\boldsymbol{x+2 < 10}[/latex]

x plus 2 is less than 10.

Exercise 3

Translate from algebra to words.

  1. [latex]14 \le 27[/latex]
  2. [latex]19-2 \ne 8[/latex]
  3. [latex]12 > 4 \div 2[/latex]
  4. [latex]x-7 < 1[/latex]

Exercise 3 Answers

  1. Fourteen is less than or equal to twenty-seven.
  2. Nineteen minus two is not equal to eight.
  3. Twelve is greater than four divided by two.
  4. x minus seven is less than one.

Exercise 4

Translate from algebra to words.

  1. [latex]19 \ge 15[/latex]
  2. [latex]7=12-5[/latex]
  3. [latex]15\div 3 < 8[/latex]
  4. [latex]y-3 > 6[/latex]

Exercise 4 Answers

  1. Nineteen is greater than or equal to fifteen.
  2. Seven is equal to twelve minus five.
  3. Fifteen divided by three is less than eight.
  4. y minus three is greater than six.

Example C

Figure 8.1.3 compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol =, < or >. in each expression to compare the fuel economy of the cars.

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This table has two rows and six columns. The first column is a header column and it labels each row The first row is labeled “Car” and the second “Fuel economy (mpg)”. To the right of the ‘Car’ row are the labels: “Prius”, “Mini Cooper”, “Toyota Corolla”, “Versa”, “Honda Fit”. Each of these columns contains an image of the labeled car model. To the right of the “Fuel economy (mpg)” row are the algebraic equations: the letter p, the equals symbol, the number forty-eight; the letter m, the equals symbol, the number twenty-seven; the letter c, the equals symbol, the number twenty-eight; the letter v, the equals symbol, the number twenty-six; and the letter f, the equals symbol, the number twenty-seven.
Figure 8.1.3: (credit: modification of work by Bernard Goldbach, Wikimedia Commons)
  1. MPG of Prius_____ MPG of Mini Cooper
  2. MPG of Versa_____ MPG of Fit
  3. MPG of Mini Cooper_____ MPG of Fit
  4. MPG of Corolla_____ MPG of Versa
  5. MPG of Corolla_____ MPG of Prius

 

a. MPG of Prius_____ MPG of Mini Cooper

Step 1: Find the values in the chart.

48____27

Step 2: Compare.

48____27

MPG of Prius > MPG of Mini Cooper

 

b. MPG of Versa_____ MPG of Fit

Step 1: Find the values in the chart.

26____27

Step 2: Compare.

26 < 27

MPG of Versa < MPG of Fit

 

c. MPG of Mini Cooper_____ MPG of Fit

Step 1: Find the values in the chart.

27____27

Step 2: Compare.

27 = 27

MPG of Mini Cooper = MPG of Fit

 

d. MPG of Corolla_____ MPG of Versa

Step 1: Find the values in the chart.

28____26

Step 2: Compare.

28 > 26

MPG of Corolla > MPG of Versa

 

e. MPG of Corolla_____ MPG of Prius

Step 1: Find the values in the chart.

28____48

Step 2: Compare.

28 < 48

MPG of Corolla < MPG of Prius

Exercise 5

Use Figure 8.1.3 to fill in the appropriate =, <, or >.

  1. MPG of Prius_____MPG of Versa
  2. MPG of Mini Cooper_____ MPG of Corolla

Exercise 5 Answers

  1. >
  2. <

Exercise 6

Use Figure 8.1.3 to fill in the appropriate =, <, or >.

  1. MPG of Fit_____ MPG of Prius
  2. MPG of Corolla _____ MPG of Fit

Exercise 6 Answers

  1. <
  2. <

Grouping symbols in algebra is much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions.

Table 8.1.4 below lists three of the most commonly used grouping symbols in algebra.

Table 8.1.4: Common Grouping Symbols
Name Symbol
parentheses [latex]\left(\phantom{\rule{0.5em}{0ex}}\right)[/latex]
brackets [latex]\left[\phantom{\rule{0.5em}{0ex}}\right][/latex]
braces [latex]\left\{\phantom{\rule{0.5em}{0ex}}\right\}[/latex]

Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

[latex]8\left(14-8\right)\phantom{\rule{4em}{0ex}}21-3\left[2+4\left(9-8\right)\right]\phantom{\rule{4em}{0ex}}24\div \left\{13-2\left[1\left(6-5\right)+4\right]\right\}[/latex]

Identify Expressions & Equations

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb.

In algebra, we have expressions and equations. An expression is like a phrase. Table 8.1.5 some examples of expressions and how they relate to word phrases.

Table 8.1.5: Expression Examples
Expression Words Phrase
[latex]3+5[/latex] 3 plus 5 The sum of three and five
[latex]n-1[/latex] n minus one The difference of n and one
[latex]6\cdot 7[/latex] 6 times 7 The product of six and seven
[latex]\dfrac{x}{y}[/latex] x divided by y The quotient of x and y

Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Table 8.1.6 shows some examples of equations.

Table 8.1.6: Equation Examples
Equation Sentence
[latex]3+5=8[/latex] The sum of three and five is equal to eight.
[latex]n-1=14[/latex] n minus one equals fourteen.
[latex]6\cdot 7=42[/latex] The product of six and seven is equal to forty-two.
[latex]x=53[/latex] x is equal to fifty-three.
[latex]y+9=2y-3[/latex] y plus nine is equal to two y minus three.

Expressions and equations:

An expression is a number, a variable, or a combination of numbers, variables, and operation symbols.

An equation is made up of two expressions connected by an equal sign.

Example D

Determine if each is an expression or an equation:

  1. [latex]16-6=10[/latex]
  2. [latex]4\cdot 2+1[/latex]
  3. [latex]x\div 25[/latex]
  4. [latex]y+8=40[/latex]

 

a. [latex]\boldsymbol{16-6=10}[/latex]

This is an equation — two expressions are connected with an equal sign.

 

b. [latex]\boldsymbol{4\cdot 2+1}[/latex]

This is an expression — no equal sign.

 

c. [latex]\boldsymbol{x\div 25}[/latex]

This is an expression — no equal sign.

 

d. [latex]\boldsymbol{y+8=40}[/latex]

This is an equation — two expressions are connected with an equal sign.

Exercise 7

For each one, determine if it is an expression or an equation.

  1. [latex]23+6=29[/latex]
  2. [latex]7\cdot 3-7[/latex]

Exercise 7 Answers

  1. equation
  2. expression

Exercise 8

For each one, determine if it is an expression or an equation.

  1. [latex]y\div 14[/latex]
  2. [latex]x-6=21[/latex]

Exercise 8 Answers

  1. expression
  2. equation

Attributions

All figures in this chapter are from 1.2 Use the Language of Algebra in Introductory Algebra by Izabela Mazur, via BCcampus.

This chapter has been adapted from 1.2 Use the Language of Algebra in Introductory Algebra (BCcampus) by Izabela Mazur (2021), which is under a CC BY 4.0 license.

The original chapter was adapted from 2.1 Use the Language of Algebra in Prealgebra 2e (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis (2020), which is under a CC BY 4.0 license. Adapted by Izabela Mazur.

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