in a problem. It’s easier to tell where a group starts and ends.
Radical : This is used for finding roots.
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 explanations and examples in Chapters 3 and 7 should clear up any questions you may have.
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. Identify the grouping symbols shown in
A. The first grouping symbol to recognize is the fraction line. It separates the term in the numerator,
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17 Write the expression using the correct symbols: “Add 2 and y; then divide that sum by 11.”
18 Identify the grouping symbols in
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.
A.
Q. Write this expression using mathematical symbols: “The circumference, C, of a circle divided by the diameter, d, is equal to pi, which is about 3.1416.”
A.
19 When you multiply the difference between z and 3 by 9, the product is equal to 13.
20 Dividing 12 by x is approximately the cube of 4.
21 The sum of y and 6 is less than the product of x and –2.
22 The square of m is greater than or equal to the square root of n.
Taking on algebraic tasks
Algebra involves symbols, such as variables and operation signs, which are the tools that you can use to make algebraic expressions more usable and readable. These things go hand in hand with simplifying, factoring, and solving problems, which are easier to solve if broken down into basic parts. Using symbols is actually much easier than wading through a bunch of words.
To simplify means to combine all that can be combined, using allowable operations, to cut down on the number of terms, and to put an expression in an easily understandable form.
To factor means to change two or more terms to just one term using multiplication. (See Chapters 11 through 13 for more on factoring.)
To solve means to find the answer. In algebra, it means to figure out what the variable stands for. You solve for the variable to create a statement that is true. (You see solving equations and inequalities in Chapters 14 through 19.)
To check your answer means to replace the variable with the number or numbers you have found when solving an equation or inequality, and show that the statement is true.
