### Key Concepts

## 1.1 Use the Language of Algebra

**Divisibility Tests**

A number is divisible by:

2 if the last digit is 0, 2, 4, 6, or 8.

3 if the sum of the digits is divisible by 3.

5 if the last digit is 5 or 0.

6 if it is divisible by both 2 and 3.

10 if it ends with 0.**How to find the prime factorization of a composite number.**

- Step 1. Find two factors whose product is the given number, and use these numbers to create two branches.
- Step 2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
- Step 3. If a factor is not prime, write it as the product of two factors and continue the process.
- Step 4. Write the composite number as the product of all the circled primes.

**How To Find the least common multiple using the prime factors method.**

- Step 1. Write each number as a product of primes.
- Step 2. List the primes of each number. Match primes vertically when possible.
- Step 3. Bring down the columns.
- Step 4. Multiply the factors.

**Equality Symbol**

$a=b$ is read “*a*is equal to*b*.”

The symbol “=” is called the equal sign.**Inequality**

**Inequality Symbols**

Inequality Symbols Words $a\ne b$ *a*is*not equal to b.*$a<b$ *a*is*less than b.*$a\le b$ *a*is*less than or equal to b.*$a>b$ *a*is*greater than b.*$a\ge b$ *a*is*greater than or equal to b.***Grouping Symbols**

$\begin{array}{cccccc}\text{Parentheses}\hfill & & & & & \left(\phantom{\rule{0.2em}{0ex}}\right)\hfill \\ \text{Brackets}\hfill & & & & & \left[\phantom{\rule{0.2em}{0ex}}\right]\hfill \\ \text{Braces}\hfill & & & & & \left\{\phantom{\rule{0.2em}{0ex}}\right\}\hfill \end{array}$**Exponential Notation**

${a}^{n}$ means multiply*a*by itself,*n*times.

The expression ${a}^{n}$ is read*a*to the ${n}^{th}$ power.**Simplify an Expression**

To simplify an expression, do all operations in the expression.**How to use the order of operations.**

- Step 1.
Parentheses and Other Grouping Symbols
- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

- Step 2.
Exponents
- Simplify all expressions with exponents.

- Step 3.
Multiplication and Division
- Perform all multiplication and division in order from left to right. These operations have equal priority.

- Step 4.
Addition and Subtraction
- Perform all addition and subtraction in order from left to right. These operations have equal priority.

- Step 1.
**How to combine like terms.**

- Step 1. Identify like terms.
- Step 2. Rearrange the expression so like terms are together.
- Step 3. Add or subtract the coefficients and keep the same variable for each group of like terms.

Operation Phrase Expression **Addition***a*plus*b*

the sum of $a$ and*b**a*increased by*b**b*more than*a*

the total of*a*and*b**b*added to*a*$a+b$ **Subtraction***a*minus $b$

the difference of*a*and*b**a*decreased by*b**b*less than*a**b*subtracted from*a*$a-b$ **Multiplication***a*times*b*

the product of $a$ and $b$

twice*a*$a\xb7b,ab,a(b),(a)(b)$

$2a$**Division***a*divided by*b*

the quotient of*a*and*b*

the ratio of*a*and*b**b*divided into*a*$a\xf7b,a\text{/}b,\frac{a}{b},b\overline{)a}$

## 1.2 Integers

**Opposite Notation**

$$\begin{array}{c}\text{\u2212}a\phantom{\rule{0.2em}{0ex}}\text{means the opposite of the number}\phantom{\rule{0.2em}{0ex}}a\hfill \\ \text{The notation}\phantom{\rule{0.2em}{0ex}}\text{\u2212}a\phantom{\rule{0.2em}{0ex}}\text{is read as \u201cthe opposite of}\phantom{\rule{0.2em}{0ex}}a\text{.\u201d}\hfill \end{array}$$**Absolute Value**

The absolute value of a number is its distance from 0 on the number line.

The absolute value of a number*n*is written as $\left|n\right|$ and $\left|n\right|\ge 0$ for all numbers.

Absolute values are always greater than or equal to zero.**Grouping Symbols**

$$\begin{array}{cccccccccc}\text{Parentheses}\hfill & & & \left(\phantom{\rule{0.2em}{0ex}}\right)\hfill & & & \text{Braces}\hfill & & & \left\{\phantom{\rule{0.2em}{0ex}}\right\}\hfill \\ \text{Brackets}\hfill & & & \left[\phantom{\rule{0.2em}{0ex}}\right]\hfill & & & \text{Absolute value}\hfill & & & \phantom{\rule{0.2em}{0ex}}\left|\phantom{\rule{0.2em}{0ex}}\right|\hfill \end{array}$$**Subtraction Property**

$\phantom{\rule{2em}{0ex}}a-b=a+\left(\text{\u2212}b\right)$

Subtracting a number is the same as adding its opposite.**Multiplication and Division of Signed Numbers**

For multiplication and division of two signed numbers:

__Same signs____Result__• Two positives Positive • Two negatives Positive

$\phantom{\rule{12.5em}{0ex}}$If the signs are the same, the result is positive.

__Different signs____Result__• Positive and negative Negative • Negative and positive Negative

$\phantom{\rule{12.2em}{0ex}}$If the signs are different, the result is negative.**Multiplication by**$\mathrm{-1}$

$\phantom{\rule{2em}{0ex}}\mathrm{-1}a=\text{\u2212}a$

Multiplying a number by $\mathrm{-1}$ gives its opposite.**How to Use Integers in Applications.**

- Step 1.
**Read**the problem. Make sure all the words and ideas are understood - Step 2.
**Identify**what we are asked to find. - Step 3.
**Write a phrase**that gives the information to find it. - Step 4.
**Translate**the phrase to an expression. - Step 5.
**Simplify**the expression. - Step 6.
**Answer**the question with a complete sentence.

- Step 1.

## 1.3 Fractions

**Equivalent Fractions Property**

If*a*,*b*, and*c*are numbers where $b\ne 0,c\ne 0,$ then

$\phantom{\rule{2em}{0ex}}\frac{a}{b}=\frac{a\xb7c}{b\xb7c}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{a\xb7c}{b\xb7c}=\frac{a}{b}.$**How to simplify a fraction.**- Step 1.
Rewrite the numerator and denominator to show the common factors.

If needed, factor the numerator and denominator into prime numbers first. - Step 2. Simplify using the Equivalent Fractions Property by dividing out common factors.
- Step 3. Multiply any remaining factors.

- Step 1.
Rewrite the numerator and denominator to show the common factors.
**Fraction Multiplication**

If*a*,*b*,*c*, and*d*are numbers where $b\ne 0,$ and $d\ne 0,$ then

$\phantom{\rule{2em}{0ex}}\frac{a}{b}\xb7\frac{c}{d}=\frac{ac}{bd}.$

To multiply fractions, multiply the numerators and multiply the denominators.**Fraction Division**

If*a*,*b*,*c*, and*d*are numbers where $b\ne 0,c\ne 0,$ and $d\ne 0,$ then

$\phantom{\rule{2em}{0ex}}\frac{a}{b}\xf7\frac{c}{d}=\frac{a}{b}\xb7\frac{d}{c}.$

To divide fractions, we multiply the first fraction by the reciprocal of the second.**Fraction Addition and Subtraction**

If*a*,*b*, and*c*are numbers where $c\ne 0,$ then

$\phantom{\rule{2em}{0ex}}\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}.$

To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.**How to add or subtract fractions.**

- Step 1.
Do they have a common denominator?
- Yes—go to step 2.
- No—rewrite each fraction with the LCD (least common denominator).
- Find the LCD.
- Change each fraction into an equivalent fraction with the LCD as its denominator.

- Step 2. Add or subtract the fractions.
- Step 3. Simplify, if possible.

- Step 1.
**How to simplify an expression with a fraction bar.**- Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator.
- Step 2. Simplify the fraction.

**Placement of Negative Sign in a Fraction**

For any positive numbers*a*and*b*,

$\phantom{\rule{2em}{0ex}}\frac{\text{\u2212}a}{b}=\frac{a}{\text{\u2212}b}=-\frac{a}{b}.$**How to simplify complex fractions.**

- Step 1. Simplify the numerator.
- Step 2. Simplify the denominator.
- Step 3. Divide the numerator by the denominator. Simplify if possible.

## 1.4 Decimals

**How to round decimals.**- Step 1. Locate the given place value and mark it with an arrow.
- Step 2. Underline the digit to the right of the place value.
- Step 3.
Is the underlined digit greater than or equal to $5?$
- Yes: add 1 to the digit in the given place value.
- No: do
__not__change the digit in the given place value

- Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.

**How to add or subtract decimals.**- Step 1. Determine the sign of the sum or difference.
- Step 2. Write the numbers so the decimal points line up vertically.
- Step 3. Use zeros as placeholders, as needed.
- Step 4. Add or subtract the numbers as if they were whole numbers. Then place the decimal point in the answer under the decimal points in the given numbers.
- Step 5. Write the sum or difference with the appropriate sign

**How to multiply decimals.**- Step 1. Determine the sign of the product.
- Step 2. Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
- Step 3. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
- Step 4. Write the product with the appropriate sign.

**How to multiply a decimal by a power of ten.**- Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
- Step 2. Add zeros at the end of the number as needed.

**How to divide decimals.**- Step 1. Determine the sign of the quotient.
- Step 2. Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places—adding zeros as needed.
- Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
- Step 4. Write the quotient with the appropriate sign.

**How to convert a decimal to a proper fraction and a fraction to a decimal.**

- Step 1. To convert a decimal to a proper fraction, determine the place value of the final digit.
- Step 2.
Write the fraction.
- numerator—the “numbers” to the right of the decimal point
- denominator—the place value corresponding to the final digit

- Step 3. To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.

**How to convert a percent to a decimal and a decimal to a percent.**- Step 1. To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.
- Step 2. To convert a decimal to a percent, move the decimal point two places to the right and then add the percent sign.

**Square Root Notation**

$\sqrt{m}$ is read “the square root of*m*.”

If $m={n}^{2},$ then $\sqrt{m}=n,$ for $n\ge 0.$

The square root of*m*, $\sqrt{m},$ is the positive number whose square is*m*.**Rational or Irrational**

If the decimal form of a number

*repeats or stops*, the number is a rational number.*does not repeat and does not stop*, the number is an irrational number.

**Real Numbers**

## 1.5 Properties of Real Numbers

Commutative PropertyWhen adding or multiplying, changing the order gives the same result$\begin{array}{cccccc}\phantom{\rule{2em}{0ex}}\text{of addition}\phantom{\rule{0.2em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}a,b\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill \phantom{\rule{9.1em}{0ex}}a+b& =\hfill & b+a\hfill \\ \phantom{\rule{2em}{0ex}}\text{of multiplication}\phantom{\rule{0.2em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}a,b\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill \phantom{\rule{9.1em}{0ex}}a\xb7b& =\hfill & b\xb7a\hfill \end{array}$ |

Associative PropertyWhen adding or multiplying, changing the grouping gives the same result.$\begin{array}{cccccc}\phantom{\rule{2em}{0ex}}\text{of addition}\phantom{\rule{0.2em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\left(a+b\right)+c& =\hfill & a+\left(b+c\right)\hfill \\ \phantom{\rule{2em}{0ex}}\text{of multiplication}\phantom{\rule{0.2em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\left(a\xb7b\right)\xb7c& =\hfill & a\xb7\left(b\xb7c\right)\hfill \end{array}$ |

Distributive Property$\begin{array}{cccccc}\phantom{\rule{2em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill \phantom{\rule{12.4em}{0ex}}a\left(b+c\right)& =\hfill & ab+ac\hfill \\ \\ & & & \hfill \phantom{\rule{12.4em}{0ex}}\left(b+c\right)a& =\hfill & ba+ca\hfill \\ \\ & & & \hfill \phantom{\rule{12.4em}{0ex}}a\left(b-c\right)& =\hfill & ab-ac\hfill \\ \\ & & & \hfill \phantom{\rule{12.4em}{0ex}}\left(b-c\right)a& =\hfill & ba-ca\hfill \end{array}$ |

Identity Property$\begin{array}{cccc}\phantom{\rule{2em}{0ex}}\text{of addition}\phantom{\rule{0.2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a\text{:}\hfill & & & \phantom{\rule{12.1em}{0ex}}a+0=a\hfill \\ \phantom{\rule{4em}{0ex}}0\phantom{\rule{0.2em}{0ex}}\text{is the}\phantom{\rule{0.2em}{0ex}}\text{additive identity}\hfill & & & \phantom{\rule{12.1em}{0ex}}0+a=a\hfill \\ \phantom{\rule{2em}{0ex}}\text{of multiplication}\phantom{\rule{0.2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a\text{:}\hfill & & & \phantom{\rule{12.65em}{0ex}}a\xb71=a\hfill \\ \phantom{\rule{4em}{0ex}}1\phantom{\rule{0.2em}{0ex}}\text{is the}\phantom{\rule{0.2em}{0ex}}\text{multiplicative identity}\hfill & & & \phantom{\rule{12.65em}{0ex}}1\xb7a=a\hfill \end{array}$ |

Inverse Property$\begin{array}{cccc}\phantom{\rule{2em}{0ex}}\text{of addition}\phantom{\rule{0.2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a,\hfill & & & \hfill \phantom{\rule{7.1em}{0ex}}a+\left(\text{\u2212}a\right)=0\\ \phantom{\rule{4em}{0ex}}\text{\u2212}a\phantom{\rule{0.2em}{0ex}}\text{is the}\phantom{\rule{0.2em}{0ex}}\text{additive inverse}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}a\hfill & & & \\ \phantom{\rule{4em}{0ex}}\text{A number and its}\phantom{\rule{0.2em}{0ex}}opposite\phantom{\rule{0.2em}{0ex}}\text{add to zero.}\hfill & & & \\ \phantom{\rule{2em}{0ex}}\text{of multiplication}\phantom{\rule{0.2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a,a\ne 0\hfill & & & \hfill \phantom{\rule{7.4em}{0ex}}a\xb7\frac{1}{a}=1\\ \\ \phantom{\rule{4em}{0ex}}\frac{1}{a}\phantom{\rule{0.2em}{0ex}}\text{is the}\phantom{\rule{0.2em}{0ex}}\text{multiplicative inverse}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}a\hfill & & & \\ \phantom{\rule{4em}{0ex}}\text{A number and its}\phantom{\rule{0.2em}{0ex}}reciprocal\phantom{\rule{0.2em}{0ex}}\text{multiply to one.}\hfill & & & \end{array}$ |

Properties of Zero$\begin{array}{cccc}\phantom{\rule{2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a,\hfill & & & \phantom{\rule{16.7em}{0ex}}a\xb70=0\hfill \\ & & & \phantom{\rule{16.7em}{0ex}}0\xb7a=0\hfill \\ \phantom{\rule{2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a,a\ne 0,\hfill & & & \phantom{\rule{17.7em}{0ex}}\frac{0}{a}=0\hfill \\ \phantom{\rule{2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a,\hfill & & & \phantom{\rule{15.7em}{0ex}}\frac{a}{0}\phantom{\rule{0.2em}{0ex}}\text{is undefined}\hfill \end{array}$ |