2.1 Use the Language of Algebra

Learning objectives

Learning objectives

• Use variables and algebraic symbols

• Identify expressions and equations

• Simplify expressions with exponents

• Simplify expressions using the order of operations

Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they 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 stays 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 $g+3$ to represent Alex’s age. See Table 2.1.

Table

2.1

Letters are used to represent variables. Letters often used for variables are $x,y,a,b,\mathrm{and}\,c.$

To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In Whole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.

In algebra, the cross symbol, $\times,$ is not used to show multiplication because that symbol may cause confusion. Does $3xy$ mean $3\,\times\,y$ (three times $y$) or $3\cdot{}x\cdot{}y$ (three times $x\,\mathrm{times}\,y$)? 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 $5+3.$

• The difference of $9$ and $2$ means subtract $9$ minus $2,$ which we write as $9-2.$

• The product of $4$ and $8$ means multiply $4$ times $8,$ which we can write as $4\cdot{}8.$

• The quotient of $20$ and $5$ means divide $20$ by $5,$ which we can write as $20\mathrm{÷}5.$

Example 2.1

Solution

1. Read ⓐ $12+14$ is "12 plus 14", or the sum of twelve and fourteen.

2. Read ⓑ $(30)(5)$ is "30 times 5", or the product of thirty and five.

3. Read ⓒ $64\mathrm{÷}8$ is "64 divided by 8", or the quotient of sixty-four and eight.

4. Read ⓓ $x-y$ is "$x$ minus $y$", or the difference of $x$ and $y$.

Takeaway: Each operation has a word name: plus or sum for addition, times or product for multiplication, divided by or quotient for division, and minus or difference for subtraction.

When two quantities have the same value, we say they are equal and connect them with an 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 that 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 $\mathrm{"<"}$ and $\mathrm{">"}$ for inequalities.

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

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

We summarize the symbols of equality and inequality in Table 2.2.

Table

2.2

Example 2.2

Solution

1. Identify the relation symbols. Each statement uses a comparison symbol, and its verbal meaning must preserve the order of the two sides.
2. Translate part ⓐ. The symbol $\le$ means less than or equal to, so $20\le{}35$ reads twenty is less than or equal to thirty-five.
3. Translate part ⓑ. The symbol $\ne$ means not equal to, so $11\ne{}15-3$ reads eleven is not equal to fifteen minus three.
4. Translate part ⓒ. The symbol $>$ means greater than, so $9>10/2$ reads nine is greater than ten divided by two.
5. Translate part ⓓ. The symbol $<$ means less than, so $x+2<10$ reads x plus two is less than ten.

Takeaway: The symbols $\le$, $\ne$, $>$, and $<$ are read less than or equal to, not equal to, greater than, and less than, keeping the order of the two sides.

Example 2.3

Solution

1. Read the chart values. The figure gives Prius $48$, Mini Cooper $27$, Versa $26$, Fit $27$, and Corolla $28$ miles per gallon.
2. Compare the Prius and Mini Cooper. Since $48>27$, the Prius has the greater fuel economy.
3. Compare the Versa and Fit. Since $26<27$, the Fit has the greater fuel economy.
4. Compare the Mini Cooper and Fit. Both show $27$ miles per gallon, so their values are equal.
5. Compare the Corolla and Versa. Since $28>26$, the Corolla has the greater fuel economy.
6. Compare the Corolla and Prius. Since $28<48$, the Prius has the greater fuel economy.

Takeaway: Compare the chart values numerically before choosing a symbol: $48>27$, $26<27$, $27=27$, $28>26$, and $28<48$.

Grouping symbols in algebra are 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 2.3 lists three of the most commonly used grouping symbols in algebra.

Table

2.3

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

Identify Expressions and 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. Here are some examples of expressions and how they relate to word phrases:

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. Here are some examples of equations:

Example 2.4

Solution

1. Recall the distinction. An equation contains an equals sign joining two expressions, while an expression contains only terms and operations.
2. Classify part ⓐ. The statement $16-6=10$ has an equals sign between two expressions, so it is an equation.
3. Classify part ⓑ. The statement $4\cdot{}2+1$ has no equals sign at all, so it is an expression.
4. Classify part ⓒ. The third statement divides $x$ by $25$ and contains a variable and an operation but no equals sign, so it is an expression.
5. Classify part ⓓ. The statement $y+8=40$ joins two expressions with an equals sign, so it is an equation.

Takeaway: An expression contains one or more terms without an equality sign, while an equation joins two expressions with an equals sign.

Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify $4\cdot{}2+1$ we’d first multiply $4\cdot{}2$ to get $8$ and then add the $1$ to get $9.$ A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

Suppose we have the expression $2\cdot{}2\cdot{}2\cdot{}2\cdot{}2\cdot{}2\cdot{}2\cdot{}2\cdot{}2.$ We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write $2\cdot{}2\cdot{}2$ as ${2}^{3}$ and $2\cdot{}2\cdot{}2\cdot{}2\cdot{}2\cdot{}2\cdot{}2\cdot{}2\cdot{}2$ as ${2}^{9}.$ In expressions such as ${2}^{3},$ the $2$ is called the base and the $3$ is called the exponent. The exponent tells us how many factors of the base we have to multiply.

We say ${2}^{3}$ is in exponential notation and $2\cdot{}2\cdot{}2$ is in expanded notation.

For powers of $n=2$ and $n=3,$ we have special names.

Table 2.4 lists some examples of expressions written in exponential notation.

Table

2.4

Example 2.5

Solution

1. Recall the definition. A product written in exponential form uses the repeated factor as the base and the count of repetitions as the exponent.
2. Count the factors in part ⓐ. The factor $16$ appears seven times, so the product is written as $16^7$.
3. Count the factors in part ⓑ. The factor $9$ appears five times, so the product is written as $9^5$.
4. Count the factors in part ⓒ. The factor $x$ appears four times, so the product is written as $x^4$.
5. Count the factors in part ⓓ. The factor $a$ appears eight times, so the product is written as $a^8$.

Takeaway: A repeated factor becomes the base and the number of repetitions becomes the exponent, giving $16^7$, $9^5$, $x^4$, and $a^8$.

Example 2.6

Solution

1. Read the base and exponent. In a power $b^n$, the base $b$ is the repeated factor and the exponent $n$ gives the number of repetitions.
2. Expand part ⓐ. The base $8$ must appear six times, so $8^6=8\cdot{}8\cdot{}8\cdot{}8\cdot{}8\cdot{}8$.
3. Expand part ⓑ. The base $x$ must appear five times, so $x^5=x\cdot{}x\cdot{}x\cdot{}x\cdot{}x$.
4. Check the factor count. The first expansion has six factors of $8$ and the second has five factors of $x$, matching each exponent.

Takeaway: Expanding a power writes the base multiplied by itself as many times as the exponent, so $8^6$ has six factors of $8$ and $x^5$ has five factors of $x$.

To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

Example 2.7

Solution

1. Expand the power. Since $3^4$ uses base $3$ and exponent $4$, write it as $3\cdot{}3\cdot{}3\cdot{}3$.
2. Multiply the first pair. Compute $3\cdot{}3=9$, leaving $9\cdot{}3\cdot{}3$.
3. Continue multiplying. Compute $9\cdot{}3=27$, leaving $27\cdot{}3$.
4. Finish the product. The last multiplication gives $27\cdot{}3=81$, so $3^4=81$.

Takeaway: To simplify $3^4$, expand it as four factors of $3$ and multiply from left to right, obtaining $81$.

Simplify Expressions Using the Order of Operations

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.

For example, consider the expression:

Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.

Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. Please Excuse My Dear Aunt Sally.

It’s good that ‘My Dear’ goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.

Similarly, ‘Aunt Sally’ goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.

Example 2.8

Solution

1. Apply multiplication before addition. In ⓐ, multiply $3\cdot{}7=21$, so $4+3\cdot{}7=4+21=25$.
2. Evaluate the parentheses first. In ⓑ, simplify $4+3=7$ inside the parentheses and then multiply: $(4+3)\cdot{}7=7\cdot{}7=49$.
3. Compare the two results. The figures show $4+3\cdot{}7$ with $3$ and $7$ highlighted before the calculation becomes $4+21=25$, while the second expression highlights $4+3$ before becoming $(4+3)\cdot{}7=49$. Thus the answers are ⓐ $25$ and ⓑ $49$.

Takeaway: Multiplication normally precedes addition, but parentheses always identify a group that must be simplified before multiplication is applied.

Example 2.9

Solution

1. Evaluate from left to right. In ⓐ, divide first: $18\mathrm{÷}9=2$, so $18\mathrm{÷}9\cdot{}2=2\cdot{}2=4$.
2. Keep the second order. In ⓑ, multiply first: $18\cdot{}9=162$, so $18\cdot{}9\mathrm{÷}2=162\mathrm{÷}2=81$.
3. Check the highlighted stages. The figures trace ⓐ through $18\mathrm{÷}9\cdot{}2$ to $2.2$ and then $4$, while they trace ⓑ through $162\mathrm{÷}2$ to $81$. Thus the answers are ⓐ $4$ and ⓑ $81$.

Takeaway: When multiplication and division have equal priority, evaluate them from left to right rather than choosing whichever operation seems easier first.

Example 2.10

Solution

1. Simplify the parentheses. Compute $5-2=3$, so $18\mathrm{÷}6+4(5-2)=18\mathrm{÷}6+4(3)$.
2. Multiply the grouped factor. Evaluate $4(3)=12$, giving $18\mathrm{÷}6+12$.
3. Divide next. Compute $18\mathrm{÷}6=3$, so the expression becomes $3+12$.
4. Add the remaining terms. The figures show $18\mathrm{÷}6+4(3)$, then $3+4(3)$, then $3+12$, and finally $15$. Therefore the simplified value is $15$.

Takeaway: Evaluate the innermost parentheses first, then complete multiplication and division before performing the final addition.

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

Example 2.11

Solution

1. Simplify the inner parentheses. Compute $4-2=2$, giving $5+2^3+3[6-3(2)]$.
2. Evaluate the multiplication inside the bracket. Compute $3(2)=6$, so $5+2^3+3[6-6]$.
3. Finish the bracket and evaluate the power. The bracket becomes $0$, while $2^3=8$, giving $5+8+3[0]$.
4. Complete the remaining products and additions. Multiply $3[0]=0$, then add to obtain $5+8+0=13$. The figures highlight these stages from the inner subtraction through the final value, so the simplified result is $13$.

Takeaway: Working from the innermost grouping outward keeps each substitution small and prevents signs, powers, or products from being misplaced.

Example 2.12

Solution

1. Evaluate the powers. Since $2^3=8$, $3^4=81$, and $5^2=25$, the expression becomes $8+81\mathrm{÷}3-25$.
2. Perform the division. Compute $81\mathrm{÷}3=27$, giving $8+27-25$.
3. Add and subtract in order. Add $8+27=35$, then subtract $25$ to obtain $35-25=10$. The figures show these highlighted stages and the final value $10$, so the simplified expression equals $10$.

Takeaway: Replace powers and complete division before carrying out addition and subtraction, which preserves the required order of operations.

Section 2.1 Exercises

Section 2.1 Exercises

Use Variables and Algebraic Symbols

In the following exercises, translate from algebraic notation to words.

Identify Expressions and Equations

In the following exercises, determine if each is an expression or an equation.

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

In the following exercises, write in expanded form.

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

1. Translate from algebra to words.

1. ⓐ $\,18+11\,$

2. ⓑ $\,(27)(9)\,$

3. ⓒ $\,84\mathrm{÷}7$

4. ⓓ $p-q$

2. Translate from algebra to words.

1. ⓐ $\,47-19\,$

2. ⓑ $\,72\mathrm{÷}9\,$

3. ⓒ $\,m+n\,$

4. ⓓ $\,(13)(7)$

3. Translate from algebra to words.

1. ⓐ $\,14\le{}27\,$

2. ⓑ $\,19-2\ne{}8\,$

3. ⓒ $\,12>4\mathrm{÷}2\,$

4. ⓓ $\,x-7<1$

1. Translate from algebra to words.

1. ⓐ $\,19\ge{}15\,$

2. ⓑ $\,7=12-5\,$

3. ⓒ $\,15\mathrm{÷}3<8\,$

4. ⓓ $\,y-3>6$

2. Use Figure 2.2 to fill in the appropriate $\mathrm{symbol},=,<,\mathrm{or}\,>.$

1. ⓐ MPG of Prius\_\_\_\_\_MPG of Versa

2. ⓑ MPG of Mini Cooper\_\_\_\_\_ MPG of Corolla

3. Use Figure 2.2 to fill in the appropriate $\mathrm{symbol},=,<,\mathrm{or}\,>.$

1. ⓐ MPG of Fit\_\_\_\_\_ MPG of Prius

2. ⓑ MPG of Corolla \_\_\_\_\_ MPG of Fit

1. Determine if each is an expression or an equation:

1. ⓐ $\,23+6=29\,$

2. ⓑ $\,7\cdot{}3-7$

2. Determine if each is an expression or an equation:

1. $\,y\mathrm{÷}14\,$

2. $\,x-6=21$

3. Write each expression in exponential form:

$41\cdot{}41\cdot{}41\cdot{}41\cdot{}41$

1. Write each expression in exponential form:

$7\cdot{}7\cdot{}7\cdot{}7\cdot{}7\cdot{}7\cdot{}7\cdot{}7\cdot{}7$

2. Write each exponential expression in expanded form:

1. ⓐ $\,{4}^{8}$

2. ⓑ $\,{a}^{7}$

3. Write each exponential expression in expanded form:

1. ⓐ ${\,8}^{8}\,$

2. ⓑ $\,{b}^{6}$

1. Simplify:

1. ⓐ $\,{5}^{3}\,$

2. ⓑ $\,{1}^{7}$

2. Simplify:

1. ⓐ $\,{7}^{2}\,$

2. ⓑ $\,{0}^{5}$

3. Simplify the expressions:

1. ⓐ $\,12-5\cdot{}2\,$

2. ⓑ $\,(12-5)\cdot{}2$

1. Simplify the expressions:

1. ⓐ $\,8+3\cdot{}9\,$

2. ⓑ $(8+3)\cdot{}9$

2. $16-9$

3. $5\cdot{}6$

1. $28\mathrm{÷}4$

2. $x+8$

3. $(2)(7)$

1. $14<21$

2. $36\ge{}19$

3. $3n=24$

1. $y-1>6$

2. $2\le{}18\mathrm{÷}6$

3. $a\ne{}7\cdot{}4$

1. $9\cdot{}6=54$

2. $5\cdot{}4+3$

3. $x+7$

1. $y-5=25$

2. $3\cdot{}3\cdot{}3\cdot{}3\cdot{}3\cdot{}3\cdot{}3$

3. $x\cdot{}x\cdot{}x\cdot{}x\cdot{}x$

1. ${5}^{3}$

2. ${2}^{8}$

3. 1. ⓐ $\,3+8\cdot{}5\,\,$

2. ⓑ $\,\mathrm{(3+8)}\cdot{}5$