APPENDIX | Algebra Review

Algebra Review I: Handling Operations

A. Order of Operations

When expressions have more than one operation, we need to know the order in which we perform the operations. Here are the rules for the order of operations:

Perform operations within parentheses (or brackets), working from the innermost out.
Evaluate exponents or roots.
Do any multiplication or division, going from left to right.
Do any addition or subtraction, going from left to right.

Most calculators (scientific calculators and graphing calculators) follow the order of operations given above.

Example 1. Evaluate the expression $2 \times 6 - 36 \div {(7 - 4)}^{2}$ .

$\begin{array}{l} 2 \times 6 - 36 \div \underset{Start here}{\underset{︸}{{(7 - 4)}^{2}}} & Perform the operation within the parentheses . \\ = 2 \times 6 - 36 \div {(3)}^{2} & Perform the square . \\ = 2 \times 6 - 36 \div 9 & Perform the multiplication and division from left to right . \\ = 12 - 4 & Finally, perform the subtraction . \\ = 8 \end{array}$

When a fraction is involved, we treat the numerator and denominator as each being enclosed in parentheses. Perform the calculation in the numerator and then perform the calculation in the denominator. Finish the problem by performing the division.

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Example 2. Use order of operations to simplify $\begin{array}{l} \frac{128 - 4 (2 + 14)}{7 + 5^{2}} . \\ \frac{128 - 4 (2 + 14)}{7 + 5^{2}} & Working on the numerator, perform the calculation in the parentheses . \\ = \frac{128 - 4 (16)}{{7+5}^{2}} & Continuing with the numerator, perform the multiplication . \\ = \frac{128 - 64}{7 + 5^{2}} & Finishing with the numerator, perform the subtraction . \\ = \frac{64}{7 + 5^{2}} & Switching to the denominator, square 5 to get 25 . \\ = \frac{64}{7+25} & Finishing with the denominator, compute the sum . \\ = \frac{64}{32} & Perform the division . \\ =2 \end{array}$

Note: When evaluating expressions such as $- 24$ , treat the expression as $- 1 \cdot 2^{4}$ . In this case, you would perform the power first and then the multiplication:

$- 2^{4} = - 1 \cdot 2^{4} = - 1 \cdot 16 = - 16$

Practice Exercises

Use order of operations to simplify each of the following by hand. Leave any fractions as fractions, without computing a decimal equivalent. If you have a scientific or graphing calculator, check your answers using your calculator.

$(5) (6) - 10 \div 2 + 1$
$5 + 44 - (6 + 8 \div 2) + (2) (3)$
$- 32 + 2 \cdot 3 + 10$
$5 - {(2 + 3)}^{2} + (8) (2) \div 4$
$\frac{5 - 4}{7 - (- 2)}$
$\frac{5 - (- 3)}{- 2 - (- 7)}$
$\frac{(- 5) (3) + 4^{2}}{6 + 3^{2}}$
$\frac{15 \div 5 \cdot 4 \div 6 - 8}{- 6 - (- 5) - 8 \div 2}$

B. Distributive Law

In expressions involving multiplication and addition/subtraction, you can distribute the multiplication over the addition/subtraction. Here is a statement of the distributive law:

$a (b + c) = a b + a c or a (b - c) = a b - a c$

Example 1. Use the distributive law to remove the parentheses in the expression $3 (2 x + 5)$ .

In the expression $3 (2 x + 5), a = 3, b = 2 x$ , and $c = 5$ . Applying the distributive property gives:

$3 (2 x + 5) = (3) (2 x) + (3) (5) = 6 x + 15$

Example 2. Apply the distributive law: $- 2 (x - 3)$ .

$- 2 (x - 3) = (- 2) (x) - (- 2) (3) = - 2 x + 6$

Factoring an expression means writing it as a product of factors, or “unmultiplying” it. One way to factor an expression is to check to see whether the terms have a factor in common. If so, use the distributive law (in reverse) as follows:

$a b + a c = a (b + c) or a b - a c = a (b - c)$

Example 3. Factor $2 x + 4$ .

Both terms have a common factor of 2. So we factor out the 2 as follows:

$4 x + 2 = (2) (2 x) + (2) (1) = 2 (2 x + 1)$

Example 4. Factor $6 x^{2} - 9 x$ .

Both terms have a common factor of $3 x$ . So we factor out the $3 x$ as follows:

$6 x^{2} - 9 x = (3 x) (2 x) - (3 x) (3) = 3 x (2 x - 3)$

Practice Exercises

In Practice Exercises 1–5, apply the distributive law to remove the parentheses.

$2 (x + 1)$
$7 (2 w - 3)$
$(x + 4) (- 3)$
$3.1 (2 p + 1.5)$
$5 (2 x + y - 3)$

In Practice Exercises 6–8, use the distributive law to factor.

$12 x - 6$
$24 x^{2} + 10 x$
$2 + 4 x + 8 x^{2}$

C. Operations with Rational Numbers (Fractions)

Rational numbers (fractions) are numbers of the form $\frac{m}{n}$ where $m$ and $n (n \neq 0)$ are integers.

Adding and Subtracting Fractions

To add or subtract fractions, convert all the individual fractions to equivalent fractions that share the same denominator. Then add or subtract numerators and place the result over the shared denominator. The lowest common denominator (or LCD) is the simplest denominator that fractions have in common. Using the lowest common denominator usually makes the work easier.

Example 1. Subtract $\frac{1}{6}$ from $\frac{3}{4}$ .

The denominators are $6 = 2 \cdot 3$ and $4 = 2 \cdot 2$ . The lowest common denominator is $2 \cdot 2 \cdot 3 = 12$ .

$\begin{array}{l} \frac{3}{4} - \frac{5}{6} = \frac{3 \cdot 3}{4 \cdot 3} - \frac{5 \cdot 2}{6 \cdot 2} & Replace each fraction with an equivalent fraction with a denominator of 12 . \\ \frac{3 \cdot 3}{4 \cdot 3} - \frac{5 \cdot 2}{6 \cdot 2} = \frac{9}{12} - \frac{10}{12} = \frac{9 - 10}{12} = \frac{- 1}{12} & Since both fractions have the same denominator, subtract the numerators . \\ \frac{- 1}{12} = \frac{- 1}{12} & It doesn't matter if the "_" is in the numerator or out in front of the entire fraction . \end{array}$

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Multiplying Fractions

To multiply two fractions, multiply the numerators and multiply the denominators. You want to cancel any common factors before carrying out the multiplication, so write the numerators and denominators in factored form.

Example 2. Multiply $\frac{15}{49}$ by $\frac{14}{25}$ .

$\frac{15}{49} \times \frac{14}{25} = \frac{15 \times 14}{49 \times 25} = \frac{(5 .3) \times (2. 7)}{(7. 7) \times (5 .5)} = \frac{6}{35}$

Dividing Fractions

To divide any expression by a fraction, multiply the expression by the reciprocal of the fraction.

Example 3. Divide 5 by $\frac{1}{6}$ .

$\frac{5}{\frac{1}{6}} = 5.6 = 30$

Example 4. Determine $\frac{1}{2} \div \frac{3}{4}$ .

$\frac{1}{2} \div \frac{3}{4} = \frac{\frac{1}{2}}{\frac{3}{4}} = \frac{1}{2} \cdot \frac{4}{3} = \frac{1 \cdot 4}{2 \cdot 3} = \frac{2 \cdot 2}{2 \cdot 3} = \frac{2}{3}$

Practice Exercises

Determine the values of the following. Express your answers in reduced form.

$\frac{6}{35} - \frac{3}{20}$
$(\frac{1}{9}) (\frac{3}{4})$
$\frac{\frac{4}{9}}{5}$
$\frac{\frac{4}{5}}{\frac{2}{3}}$
$\frac{\frac{2}{5} + \frac{1}{10}}{2}$
$\frac{1}{3} (\frac{6}{5} - \frac{3}{10})$
$\frac{2}{15} + \frac{6}{25}$
$\frac{\frac{2}{3}}{\frac{3}{4} + \frac{4}{5}}$