has two factors, the 2 and the z. So there are a total of five factors.
3. The variables are h, b1, and b2; the constant is the fraction
4. 2,
In algebra today, a variable represents the unknown. (You can see more on variables in the “Speaking in Algebra” section earlier in this chapter.) Before the use of symbols caught on, problems were written out in long, wordy expressions. Actually, using letters, signs, and operations was a huge breakthrough. First, a few operations were used, and then algebra became fully symbolic. Nowadays, you may see some words alongside the operations to explain and help you understand, like having subtitles in a movie.
By doing what early mathematicians did – letting a variable represent a value, then throwing in some operations (addition, subtraction, multiplication, and division), and then using some specific rules that have been established over the years – you have a solid, organized system for simplifying, solving, comparing, or confirming an equation. That’s what algebra is all about: That’s what algebra’s good for.
Deciphering the symbols
The basics of algebra involve symbols. Algebra uses symbols for quantities, operations, relations, or grouping. The symbols are shorthand and are much more efficient than writing out the words or meanings. But you need to know what the symbols represent, and the following list shares some of that info. The operations are covered thoroughly in Chapter 5.
✓ + means add or find the sum or more than or increased by; the result of addition is the sum. It also is used to indicate a positive number.
✓ – means subtract or minus or decreased by or less; the result is the difference. It’s also used to indicate a negative number.
✓ × means multiply or times. The values being multiplied together are the multipliers or factors; the result is the product. Some other symbols meaning multiply can be grouping symbols: ( ), [ ], { }, ·, *. In algebra, the × symbol is used infrequently because it can be confused with the variable x. The × symbol is popular because it’s easy to write. The grouping symbols are used when you need to contain many terms or a messy expression. By themselves, the grouping symbols don’t mean to multiply, but if you put a value in front of or behind a grouping symbol, it means to multiply.
✓ ÷ means divide. The number that’s going into the dividend is the divisor. The result is the quotient. Other signs that indicate division are the fraction line and slash, /.
✓
✓
✓ π is the Greek letter pi that refers to the irrational number: 3.14159… It represents the relationship between the diameter and circumference of a circle.
Grouping
When a car manufacturer puts together a car, several different things have to be done first. The engine experts have to construct the engine with all its parts. The body of the car has to be mounted onto the chassis and secured, too. Other car assemblers have to perform the tasks that they specialize in as well. When these tasks are all accomplished in order, then the car can be put together. The same thing is true in algebra. You have to do what’s inside the grouping symbol before you can use the result in the rest of the equation.
Grouping symbols tell you that you have to deal with the terms inside the grouping symbols before you deal with the larger problem. If the problem contains grouped items, do what’s inside a grouping symbol first, and then follow the order of operations. The grouping symbols are
✓ Parentheses ( ): Parentheses are the most commonly used symbols for grouping.
✓ Brackets [] and braces { }: Brackets and braces are also used frequently for grouping and have the same effect as parentheses. Using the different types of symbols helps when there’s more than one grouping in a problem. It’s easier to tell where a group starts and ends.
✓ Radical
✓ Fraction line (called the vinculum): The fraction line also acts as a grouping symbol – everything above the line (in the numerator) is grouped together, and everything below the line (in the denominator) is grouped together.
Even though the order of operations and grouping-symbol rules are fairly straightforward, it’s hard to describe, in words, all the situations that can come up in these problems. The examples in Chapters 3 and 7 should clear up any questions you may have.
Examples
Q. What are the operations found in the expression:
A. The operations, in order from left to right, are multiplication, subtraction, division, addition, multiplication, and square root. The term 3y means to multiply 3 times y. The subtraction symbol separates the first and second terms. Writing y over 4 in a fraction means to divide. Then that term has the radical added to it. The 2 and y are multiplied under the radical, and then the square root is taken.
Q. Write the expression using the correct symbols: The absolute value of the difference between x and 6 is multiplied by 7.
A. The difference between two values is the result of subtraction, so write x – 6. The absolute value of that difference is written
Practice Questions
Write the expression using the correct symbols.
1.
