Algebraic Expressions

The subject of arithmetic concerns itself with numbers and operations. Our study of algebra begins with a review of these concepts. While we shall not usually make an issue of the difference between a number and a numeral, it is appropriate to consider briefly this difference.

A number is an abstract, intangible, but carefully defined idea. Like honesty and red it is something we talk about, but cannot actually see or handle. A numeral is a symbol which we use to represent the idea of number, just as we use symbols, the words “honesty” and “red,” to represent these ideas. The number six can be represented by the numeral 4 + 2 as well as by other numerals, such as12/2, 9 -3, VI, 6, 2∙3, etc. None of these symbols is the number six ; all of them only represent the number six. Indeed, every number can be represented by many different numerals.

Some numerals include operational symbols. Thus, the representation of six by 9-3 includes numerals which represent the numbers nine and three as well as the operational symbol - indicating subtraction. For many of the numbers we shall consider there is a unique numeral called the simplest form of that number. For the natural numbers the simplest form will be a numeral without operational symbols; 1, 2, 3, 4, and so on. For numbers which must be represented by fractions, their simplest forms will be fractions in lowest terms.

A numerical expression will be a numeral which may include both number and operational symbols. We shall frequently wish to simplify numerical expressions to find the simplest form of the number they represent. When doing so, we must pay careful attention, not only to the numbers and operations indicated, but also to the order in which those operations are performed. For example, the numerical expression 4+ 7∙2 would represent the number twenty-two if the operation of addition were performed first, but would represent the number eighteen if the multiplication were performed first. In order to avoid such ambiguous situations, we shall adopt a hierarchy or order of operations.

The operation order we shall use is one which is rather generally agreed upon: multiplications and divisions will precede additions and subtractions. Thus, in simplest form, 4+7∙2=18; 24÷8+4=7, 10-3∙2=4; 15+6/3=17. In order to be able to write numerical expressions such as 4 + 7∙2 in which we wish the operation of addition to precede that of multiplication, we introduce symbols of grouping. These will usually be parentheses, (), but we shall also use brackets, [ ], or the bar, ___. Hence, in simplest form, (4+7)∙2=22; 24÷[8+4]=2; (15+6)/3=7.

While we have placed multiplication and division on a higher order level than addition and subtraction, we shall not establish a hierarchy between those operations on the same level. Thus, when only additions and subtractions appear in an expression, we agree to perform the operations as encountered from left to right, unless grouping symbols direct otherwise. Then 4+7-2+3=12; 5+9-8+1=7; 5+(6-3)+7-4=11; 14-(5+3)+9-2=13. Similarly, let us move from left to right when only multiplications and divisions appear. Then 4∙7/2=14; 50÷(2∙5)=5; 8/4∙5=10; 5∙(12/4)=15.

We may now summarize our conventions concerning the order of operations.

First Perform the operations within symbols of grouping in the prescribed order.

Second Multiply and divide from left to right.

Third Add and subtract from left to right.

Some examples will illustrate the conventions.

Example 1 Find the simplest form of 2(3+4)-5+1.

Solution 2(3 + 4)-5 +1=2∙7-5+1=14-5+1=10

Example 2 Find the simplest form of 3/4÷[1/2+3∙1/4]+1

Solution 3/4÷[1/2+3∙1/4]+1=3/4÷[1/2+3/4]+1

= 3/4÷[2/4+3/4]+1

= 3/4÷5/4+1= 3/4∙4/5+1

= 3/5+1=8/5

In a numerical expression there is no difficulty finding the simplest form of the number represented. But consider the expression 2∙x+3 . Since it is not clear which number the letter x represents, it is also not clear which number the expression 2∙x+3represents. Because we shall frequently encounter situations similar to this in our study of algebra, let us establish a vocabulary and certain conventions to cover such occurrences.

Whenever a letter is used in an expression, let us call that expression an algebraic expression. Thus 4+x-3, 12∙x-19 and 3∙x+2∙x are all algebraic expressions. Without further information, the number each algebraic expression represents cannot be determined. If in the above algebraic expressions the letter x is replaced by the numeral 3, the simplest forms of the resulting numerical expressions can be easily found.

4+x-3 becomes 4+2-3=3
12∙x-19 becomes 12∙3-19=17
3∙x+2∙x becomes 3∙3+2∙3=15
If, on the other hand, we replace x by 2, then
4+x-3 becomes 4+x-3
12∙x-19 becomes 12∙2-19=5
3∙x+2∙x becomes 3∙2+2∙2=10
Clearly the number represented by an algebraic expression depends on the replacement chosen for the letter which appears in it.

Both letters and numerals are placeholders—symbols which take the place of numbers. When a placeholder can represent only a single number we shall call it a constant. Thus, 4, 19, 21/2 are constants; x is also a constant if only one number is allowed as a replacement for it. But often any member of a set of numbers is allowed as a replacement for a letter. In that case we shall call the letter a variable. We shall call the set of replacements for any letter its replacement set. Thus, a variable is a placeholder whose replacement set contains more than one number; a constant is a placeholder which can represent only one number. For example, if the replacement set for x is {2, 3, 4}, then in the expression 3∙x-4 , x is a variable and 3 and 4 are constants. If the replacement set for x is {5}, then in the expression 2+5∙x, 2, 5, and x are all constants.

The value of an expression is the number it represents. We shall usually express the value in simplest form. Determining the simplest form of the value of a numerical expression has already been explained. But to determine the value of an algebraic expression we must first know what replacement to make for all placeholders. Hence the expression 4∙x+5 has the value 33, if the replacement for x is 7, but has the value 45 if x is replaced by 10. If we do not know the replacement for x, we cannot determine the value of 4∙x+5.

Several conventions will be followed. In an algebraic expression, symbols of multiplication will often be dropped, if one factor is a literal placeholder. We shall write 4x+5 instead of 4∙x+5. For any algebraic expression the replacement set for the variable will also be called the domain of the expression. The set of all numbers that the expression can represent with the possible replacements of the variable—the set of all values of the expression—will be called the range of the expression. Hence, if the replacement set for x is {2, 3, 4}, the domain of 4x+5 is also {2, 3, 4}, and the range of 4x+5 is {13, 17, 21}. Note that both the domain and range of an expression are sets of numbers.

Example 3 Find the value of the expression 5+2(x+1)-3 when the placeholder is replaced by 1/2.

Solution 5+2(x+1)-3 becomes 5+2(1/2+1)-3=5+2(3/2)-3=5+3-3=5. (Note that the value is 5, a number, not a set of numbers. )

We shall often use the symbolic phrase "x=7" to represent "x is replaced by 7" when the context of a problem makes it clear that this is our intention. The following examples will illustrate this notation.

Example 4 The domain of the expression (2x+3)/x is {2,1/3}. Find its range.

Solution If x=2 then

(2x+3)/x=(2∙2+3)/3=(4+3)/2=7/2

If x=1/3, then

(2x+3)/x=(2(1/3)+3)/(1/3)=(2/3+3)/(1/3)=(11/3)/(1/3)=11

The range is {7/2,11}.

Example 5 The range of the expression 2x+1 is {2}. Find its domain.

Solution We want to find a replacement for x which makes the expression 2x+1 have the value 2. Since 2(1/2)+1=1+1=2, we see that 1/2 is such a replacement. That is when x=1/2, then 2x+1=2(1/2)+1=2. Hence {1/2} is the domain of the expression.

Sometimes we shall want to represent verbal descriptions by algebraic expressions. The skill needed to make such translations is acquired gradually. Some examples will illustrate the technique involved.

Example 6 It is estimated that a total of 16 pounds of cooking equipment are needed for a four-man group which plans to backpack into the mountains. Each man must carry six pounds of extra clothing for the trip and two pounds of food for each day of the trip. What is the total weight of all equipment and supplies needed for a trip of x days?

Solution 16+4(6+2x)

Open Sentences

We have agreed to call certain collections of number, operation, and grouping symbols numerical expressions. For example, 4+(7-2/3)9, 5∙8-3∙7+11 and (2+3∙5)/(12-2)are all numerical expressions. Is 15-6∙2=7-(6-2) a numerical expression? Note that 15-6∙2 and 7-(6-2) are both numerical expressions. But because of the inclusion of the symbol =, we shall not call 15-6∙2 = 7-(6-2) a numerical expression. Similarly, we shall not call 3+7!=12-6 a numerical expression (even though it contains two such expressions), because it contains the symbol ≠ (is not equal to).

When we write 2 + 2 = 4, we mean that 2 + 2 and 4 are numerals for the same number. But clearly 10 and 6 are not numerals for the same number. Hence, when we write 3+7 = 12-6, we are making a false numerical statement. On the other hand both 15-6∙2 and 7-(6-2) are numerals which represent the number whose simplest form is 3, and so 15-6∙2 =7-(6-2) is a true numerical statement. Another example of a numerical statement is 4+7∙3 < 33+ 0. This statement includes the symbol <(is less than), and it is a true numerical statement, since it is true that 25 < 33.

A numerical statement, then, contains two numerical expressions and a connective, such as =, <, or > (is greater than), to which one of the labels “true’’ or ‘‘false” can be assigned. Deciding whether a numerical statement is true or false usually involves finding the simplest form of each numerical expression and making the indicated comparison.

Example 1 Is the following a true or false numerical statement?
4/6+9>11-4∙2
Solution True, since 29/3>3.
Example 2 Is the following a true or false numerical statement?
5+3∙2!=4∙3-1
Solution False, since the statement says that 11!=11.

For the two reasons stated below, 2+x=12-3 is not a numerical statement. First, note that on the left we have an algebraic, rather than a numerical, expression. Second, we cannot say whether this is a true or false assertion since we do not know what number to replace x by, and so do not know the value of the algebraic expression. But suppose we decide to replace x by 7. We then have the numerical statement 2 + 7 =12-3, which is true. Suppose, on the other hand, we replace x by 10. We now have 2+10=12-3, which is false. Similarly, replacing x by 41 yields another false numerical statement 2+41 = 12-3. Indeed, each replacement for x yields a numerical statement which is either true or false. But we cannot say that 2+x=12-3 is true, nor can we say that it is false. We shall call 2+x=12-3 an open sentence.

Another example of an open sentence is 2x-3 > x+7. Here both the left and right sides of the sentence are algebraic expressions, and the connective is >. But again, we cannot decide whether the sentence is true or false unless we know the replacement for x. If x is replaced by 3, the resulting numerical statement is false. If x is replaced by 12, the resulting statement is true. This is typical of open sentences. When a replacement is chosen for the variable, the open sentence becomes a numerical statement, which is either true or false.

Example 3 If the variable y is replaced by 4 in the open sentence 2y-3>y+7, is the resulting numerical statement true or false?

Solution If y = 4, then 2y-3=2∙4-3=5 and y+7=4+7=11. Then 5 > 11 is a false numerical statement.

Example 4 If the variable t is replaced by 1/2 in the open sentence 12+t=14/t, is the resulting numerical statement true or false?

Solution If t=1/2, the sentence 12t+1=14/t becomes 12(1/2)+1=14/(1/2) or 7 = 28, a false statement.

Example 5 If the variable z is replaced by 0 in the open sentence 12-z=z+12 , is the resulting statement true or false?

Solution If z = 0, then 12-z=z+12 becomes 12-0 =0+12, or 12=12, a true numerical statement.

2x + 1 is an example of an algebraic expression. Associated with the variable x is a certain replacement set which we also call the domain of the expression. If the replacement set for x in the above expression is {1, 2, 6}, then the range of the expression is {3, 5, 13}. Consider the open sentence 2x+1=5. It is not possible to assign either label true or false to this open sentence. But if we consider {1, 2, 6} as the replacement set for the variable x, then for one member of the replacement set, namely 2, the sentence 2x+1=5 becomes 2∙2+1=5, a true numerical statement. But for the other members of the replacement set, namely 1 and 6, the sentence becomes 2∙1+1=5and 2∙6+1=5, both false numerical statements. This illustrates an important property of an open sentence. For some replacements of the variable the sentence becomes a true statement, for others a false statement. Let us call the set of all members of the replacement set which result in a true statement the solution set of the open sentence. Then if the replacement set is{1, 2, 6} for the open sentence2x+1=5, we shall say the solution set is{2}.

Example 6 If the replacement set for t is {1, 4, 5, 7}, find the solution set for the open sentence (t-1)(t+1)=48.

Solution (t-1)(t+1)=48
If t = 1, (1-1)(1+1)=48 is false.
If t=3, (3-1)(3+1)=48 is false.
If t= 5, (5-1)(5+1)=48 is false.
If t = 7, (7-1)(7+1)=48 is true.
The solution set is {7}.

Example 7 If the replacement set for v is {0, 1, 2}, find the solution set for 2v+4=2(v+ 2).

Solution 2v+4=2(v+2)
If v=0, 2∙0+4=2(0+2) is true.
If v=1, 2∙1+4=2(1+2) is true.
If v=2, 2∙2+4=2(2+2) is true.
The solution set is the entire replacement set, {0, 1, 2}.

Example 8 If the replacement set for x is {0, 1}, find the solution set for x+4< 2x+1.

Solution x+4< 2x+1

If x=0, 0+4<2∙0+1 is false.
If x=1, 1+4<2∙1+1 is false.
The solution set must be empty, so we indicate this by writing { }.

In Example 7 we considered the open sentence 2v+4=2(v+2) and the replacement set {0, 1, 2}. We saw that the solution set was the entire replacement set. That is, for each replacement of v by a member of its replacement set the value of the algebraic expression 2v+4 was the same as the value of the algebraic expression 2(v+2). We shall describe this situation by saying that 2v+4 and 2(v+2) are equivalent algebraic expressions over the domain {0, 1, 2}. Consider the expressions x/2+1 and (x+2)/2. Let us use {10,20} as the replacement set for the variable. When x is replaced by 10, the first expression becomes 10/2+1=6, and the second becomes ((10+ 2))/2=6. And when x is replaced by 20, the first becomes 20/2+1=11 and the second ((20+2))/2=11. The value of each expression is the same whenever we replace the variable by the same member of its replacement set. Thus, x/2+1 and ((x+2))/2 are equivalent algebraic expressions over the domain{10, 20}.

Example 9 Are 5x+2 and (6+x)/xand equivalent algebraic expressions over the domain {1, 2, 3}?

Solution If x=1, then 5x+2=5∙1+2=7, and (6+x)/x=(6+1)/1=7
If x=2, then 5x+2=12, but (6+x)/x=8/2=4
Since 12!=4, we see that the expressions are not equivalent over the domain {1, 2,3}. Note that it is not necessary to even consider 3 as a replacement since, regardless of what the values of the expressions would be with this replacement, the expressions cannot be equivalent over a domain which includes 2.

We have seen that it is possible to form algebraic expressions from certain verbal expressions. Some verbal statements lead to open sentences involving one or more algebraic expressions. The following examples will illustrate this.

Example 10 Form an open sentence from the statement, ‘‘The number which is one more than x is 7 less than twice x.”

Solution One possibility is x+1=2x-7.

Example 11 Form an open sentence from the statement, “If the cost of a package of chewing gum increases 1 cent, the new price will be 120% of the original.”

Solution x+1=1.2x (120% = 1.2)

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