✓ × 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. The square root of x is subtracted from 3 times y.
2. Add 2 and y; then divide that sum by 11.
Practice Answers
1.
2.
Defining relationships
Algebra is all about relationships – not the he-loves-me-he-loves-me-not kind of relationship – but the relationships between numbers or among the terms of an expression. Although algebraic relationships can be just as complicated as romantic ones, you have a better chance of understanding an algebraic relationship. The symbols for the relationships are given here. The equations are found in Chapters 14 through 18, and inequalities are found in Chapter 19.
✓ = means that the first value is equal to or the same as the value that follows.
✓ ≠ means that the first value is not equal to the value that follows.
✓ means that one value is approximately the same or about the same as the value that follows; this is used when rounding numbers.
✓ ≤ means that the first value is less than or equal to the value that follows.
✓ < means that the first value is less than the value that follows.
✓ ≥ means that the first value is greater than or equal to the value that follows.
✓ > means that the first value is greater than the value that follows.
Practice Questions
Write the expression using the correct symbols.
1. When you multiply the difference between z and 3 by 9, the product is equal to 13.
2. Dividing 12 by x is approximately the cube of 4.
3. The sum of y and 6 is less than the product of x and –2.
4. The square of m is greater than or equal to the square root of n.
Practice Answers
1. (z – 3)9 = 13 or 9(z – 3) = 13. The 9 can be written behind