Chapter 1 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

##

Worked example

absolute value

additive identity

The number 0 is the additive identity because adding 0 to any number does not change its value.

additive inverse

The opposite of a number is its additive inverse.

coefficient

The coefficient of a term is the constant that multiplies the variable in a term.

complex fraction

A fraction in which the numerator or the denominator is a fraction is called a complex fraction.

composite number

A composite number is a counting number that is not prime. It has factors other than 1 and the number itself.

constant

A constant is a number whose value always stays the same.

denominator

b

divisible by a number

m

n

m

n

equation

An equation is two expressions connected by an equal sign.

equivalent fractions

Equivalent fractions are fractions that have the same value.

evaluate an expression

To evaluate an expression means to find the value of the expression when the variables are replaced by given numbers.

expression

An expression is a number, a variable, or a combination of numbers and variables using operation symbols.

factors

a

b

m

fraction

a

b

integers

The whole numbers and their opposites are called the integers.

irrational number

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

least common denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

least common multiple

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

like terms

Terms that are either constants or have the same variables raised to the same powers are called like terms.

multiple of a number

n

n.

multiplicative identity

The number 1 is the multiplicative identity because multiplying 1 by any number does not change its value.

multiplicative inverse

The reciprocal of a number is its multiplicative inverse.

negative numbers

numerator

a

opposite

The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.

order of operations

The order of operations are established guidelines for simplifying an expression.

percent

A percent is a ratio whose denominator is 100.

prime factorization

The prime factorization of a number is the product of prime numbers that equals the number.

prime number

A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.

principal square root

The positive square root is called the principal square root.

rational number

p

q

real number

A real number is a number that is either rational or irrational.

reciprocal

The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator.

simplify an expression

To simplify an expression means to do all the math possible.

square of a number

m

n

square root of a number

n

m

term

A term is a constant, or the product of a constant and one or more variables.

variable

A variable is a letter that represents a number whose value may change.

##

Key Concepts

• 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.*Table 1.4

• Grouping Symbols $\begin{array}{lccccl} \mathrm{Parentheses} & & & & & (\,) \\ \mathrm{Brackets} & & & & & [\,] \\ \mathrm{Braces} & & & & & \mathrm{\{}\,\mathrm{\}} \\ \end{array}$

• 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.

• 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

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 · $a\cdot{}b$, $ab$, $a(b)$, $(a)(b)$, $2a$ Division · a divided by b; the quotient of a and b; the ratio of a and b; b divided into a · $a\mathrm{÷}b$, $a/b$, $\dfrac{a}{b}$, $b/a$

Table 1.5

• Opposite Notation

• 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 $|n|$ and $|n|\ge{}0$ for all numbers. Absolute values are always greater than or equal to zero.

• Grouping Symbols

• Subtraction Property $\,a-b=a+(-b)$ 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 $\,$If the signs are the same, the result is positive. Different signs Result• Positive and negative Negative • Negative and positive Negative $\,$If the signs are different, the result is negative.

• Multiplication by $-1$$\,-1a=-a$ Multiplying a number by $-1$ 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.

• Equivalent Fractions Property If a, b, and c are numbers where $b\ne{}0,c\ne{}0,$ then $\,\frac{a}{b}=\frac{a\cdot{}c}{b\cdot{}c}\,\mathrm{and}\,\frac{a\cdot{}c}{b\cdot{}c}=\frac{a}{b}.$

• 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.

• Fraction Multiplication If a, b, c, and d are numbers where $b\ne{}0,$ and $d\ne{}0,$ then $\,\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}.$ 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 $\,\frac{a}{b}\mathrm{÷}\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}.$ 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 $\,\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\,\mathrm{and}\,\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}.$ 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.

• 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, $\,\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}.$

• 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.

• 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 Figure 1.9

• Commutative PropertyWhen adding or multiplying, changing the order gives the same result $\begin{array}{lccrll} \,\mathrm{of addition}\,\mathrm{If}\,a,b\,\mathrm{are real numbers, then} & & & \,a+b & = & b+a \\ \,\mathrm{of multiplication}\,\mathrm{If}\,a,b\,\mathrm{are real numbers, then} & & & \,a\cdot{}b & = & b\cdot{}a \\ \end{array}$

• Associative PropertyWhen adding or multiplying, changing the grouping gives the same result. $\begin{array}{lccrll} \,\mathrm{of addition}\,\mathrm{If}\,a,b,\,\mathrm{and}\,c\,\mathrm{are real numbers, then} & & & \,(a+b)+c & = & a+(b+c) \\ \,\mathrm{of multiplication}\,\mathrm{If}\,a,b,\,\mathrm{and}\,c\,\mathrm{are real numbers, then} & & & \,(a\cdot{}b)\cdot{}c & = & a\cdot(b\cdot{}c) \\ \end{array}$

• Distributive Property $\begin{array}{lccrll} \,\mathrm{If}\,a,b,\,\mathrm{and}\,c\,\mathrm{are real numbers, then} & & & \,a(b+c) & = & ab+ac \\ & & & \,(b+c)a & = & ba+ca \\ & & & \,a(b-c) & = & ab-ac \\ & & & \,(b-c)a & = & ba-ca \\ \end{array}$

• Identity Property $\begin{array}{lccl} \,\mathrm{of addition}\,\mathrm{For any real number}\,a: & & & \,a+0=a \\ \,0\,\mathrm{is the}\,\mathrm{additive identity} & & & \,0+a=a \\ \,\mathrm{of multiplication}\,\mathrm{For any real number}\,a: & & & \,a\cdot{}1=a \\ \,1\,\mathrm{is the}\,\mathrm{multiplicative identity} & & & \,1\cdot{}a=a \\ \end{array}$

• Inverse Property $\begin{array}{lccr} \,\mathrm{of addition}\,\mathrm{For any real number}\,a, & & & \,a+(-a)=0 \\ \,-a\,\mathrm{is the}\,\mathrm{additive inverse}\,\mathrm{of}\,a & & & \\ \,\mathrm{A number and its}\,opposite\,\mathrm{add to zero.} & & & \\ \,\mathrm{of multiplication}\,\mathrm{For any real number}\,a,a\ne{}0 & & & \,a\cdot\frac{1}{a}=1 \\ \,\frac{1}{a}\,\mathrm{is the}\,\mathrm{multiplicative inverse}\,\mathrm{of}\,a & & & \\ \,\mathrm{A number and its}\,reciprocal\,\mathrm{multiply to one.} & & & \\ \end{array}$

• Properties of Zero $\begin{array}{lccl} \,\mathrm{For any real number}\,a, & & & \,a\cdot{}0=0 \\ & & & \,0\cdot{}a=0 \\ \,\mathrm{For any real number}\,a,a\ne{}0, & & & \,\frac{0}{a}=0 \\ \,\mathrm{For any real number}\,a, & & & \,\frac{a}{0}\,\mathrm{is undefined} \\ \end{array}$

##

Review Exercises

Identify Multiples and Factors

In the following exercises, simplify each expression.

Evaluate an Expression

In the following exercises, evaluate the following expressions.

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the following expressions by combining like terms.

Translate an English Phrase to an Algebraic Expression

In the following exercises, translate the phrases into algebraic expressions.

Simplify Expressions with Absolute Value

In the following exercise, fill in $<,>,$ or $=$ for each of the following pairs of numbers.

In the following exercises, simplify.

Add and Subtract Integers

In the following exercises, simplify each expression.

Multiply and Divide Integers

In the following exercise, multiply or divide.

Simplify and Evaluate Expressions with Integers

In the following exercises, simplify each expression.

For the following exercises, evaluate each expression.

Translate English Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

Use Integers in Applications

In the following exercise, solve.

Simplify Fractions

In the following exercises, simplify.

Multiply and Divide Fractions

In the following exercises, perform the indicated operation.

Add and Subtract Fractions

In the following exercises, perform the indicated operation.

Use the Order of Operations to Simplify Fractions

In the following exercises, simplify.

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

Round Decimals

Add and Subtract Decimals

In the following exercises, perform the indicated operation.

Multiply and Divide Decimals

In the following exercises, perform the indicated operation.

Convert Decimals, Fractions and Percents

In the following exercises, convert each decimal to a fraction.

In the following exercises, convert each fraction to a decimal.

In the following exercises, convert each decimal to a percent.

Simplify Expressions with Square Roots

In the following exercises, simplify.

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

In the following exercise, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers for each set of numbers

Locate Fractions and Decimals on the Number Line

In the following exercises, locate the numbers on a number line.

Use the Commutative and Associative Properties

In the following exercises, simplify.

Use the Properties of Identity, Inverse and Zero

In the following exercises, simplify.

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the Distributive Property.

##

In the following exercises, simplify each expression.

Key Terms

absolute value

additive identity

The number 0 is the additive identity because adding 0 to any number does not change its value.

additive inverse

The opposite of a number is its additive inverse.

coefficient

The coefficient of a term is the constant that multiplies the variable in a term.

complex fraction

A fraction in which the numerator or the denominator is a fraction is called a complex fraction.

composite number

A composite number is a counting number that is not prime. It has factors other than 1 and the number itself.

constant

A constant is a number whose value always stays the same.

denominator

b

divisible by a number

m

n

m

n

equation

An equation is two expressions connected by an equal sign.

equivalent fractions

Equivalent fractions are fractions that have the same value.

evaluate an expression

To evaluate an expression means to find the value of the expression when the variables are replaced by given numbers.

expression

An expression is a number, a variable, or a combination of numbers and variables using operation symbols.

factors

a

b

m

fraction

a

b

integers

The whole numbers and their opposites are called the integers.

irrational number

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

least common denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

least common multiple

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

like terms

Terms that are either constants or have the same variables raised to the same powers are called like terms.

multiple of a number

n

n.

multiplicative identity

The number 1 is the multiplicative identity because multiplying 1 by any number does not change its value.

multiplicative inverse

The reciprocal of a number is its multiplicative inverse.

negative numbers

numerator

a

opposite

The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.

order of operations

The order of operations are established guidelines for simplifying an expression.

percent

A percent is a ratio whose denominator is 100.

prime factorization

The prime factorization of a number is the product of prime numbers that equals the number.

prime number

A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.

principal square root

The positive square root is called the principal square root.

rational number

p

q

real number

A real number is a number that is either rational or irrational.

reciprocal

The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator.

simplify an expression

To simplify an expression means to do all the math possible.

square of a number

m

n

square root of a number

n

m

term

A term is a constant, or the product of a constant and one or more variables.

variable

A variable is a letter that represents a number whose value may change.

##

Key Concepts

• 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.*Table 1.4

• Grouping Symbols $\begin{array}{lccccl} \mathrm{Parentheses} & & & & & (\,) \\ \mathrm{Brackets} & & & & & [\,] \\ \mathrm{Braces} & & & & & \mathrm{\{}\,\mathrm{\}} \\ \end{array}$

• 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.

• 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

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 · $a\cdot{}b$, $ab$, $a(b)$, $(a)(b)$, $2a$ Division · a divided by b; the quotient of a and b; the ratio of a and b; b divided into a · $a\mathrm{÷}b$, $a/b$, $\dfrac{a}{b}$, $b/a$

Table 1.5

• Opposite Notation

• 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 $|n|$ and $|n|\ge{}0$ for all numbers. Absolute values are always greater than or equal to zero.

• Grouping Symbols

• Subtraction Property $\,a-b=a+(-b)$ 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 $\,$If the signs are the same, the result is positive. Different signs Result• Positive and negative Negative • Negative and positive Negative $\,$If the signs are different, the result is negative.

• Multiplication by $-1$$\,-1a=-a$ Multiplying a number by $-1$ 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.

• Equivalent Fractions Property If a, b, and c are numbers where $b\ne{}0,c\ne{}0,$ then $\,\frac{a}{b}=\frac{a\cdot{}c}{b\cdot{}c}\,\mathrm{and}\,\frac{a\cdot{}c}{b\cdot{}c}=\frac{a}{b}.$

• 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.

• Fraction Multiplication If a, b, c, and d are numbers where $b\ne{}0,$ and $d\ne{}0,$ then $\,\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}.$ 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 $\,\frac{a}{b}\mathrm{÷}\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}.$ 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 $\,\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\,\mathrm{and}\,\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}.$ 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.

• 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, $\,\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}.$

• 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.

• 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 Figure 1.9

• Commutative PropertyWhen adding or multiplying, changing the order gives the same result $\begin{array}{lccrll} \,\mathrm{of addition}\,\mathrm{If}\,a,b\,\mathrm{are real numbers, then} & & & \,a+b & = & b+a \\ \,\mathrm{of multiplication}\,\mathrm{If}\,a,b\,\mathrm{are real numbers, then} & & & \,a\cdot{}b & = & b\cdot{}a \\ \end{array}$

• Associative PropertyWhen adding or multiplying, changing the grouping gives the same result. $\begin{array}{lccrll} \,\mathrm{of addition}\,\mathrm{If}\,a,b,\,\mathrm{and}\,c\,\mathrm{are real numbers, then} & & & \,(a+b)+c & = & a+(b+c) \\ \,\mathrm{of multiplication}\,\mathrm{If}\,a,b,\,\mathrm{and}\,c\,\mathrm{are real numbers, then} & & & \,(a\cdot{}b)\cdot{}c & = & a\cdot(b\cdot{}c) \\ \end{array}$

• Distributive Property $\begin{array}{lccrll} \,\mathrm{If}\,a,b,\,\mathrm{and}\,c\,\mathrm{are real numbers, then} & & & \,a(b+c) & = & ab+ac \\ & & & \,(b+c)a & = & ba+ca \\ & & & \,a(b-c) & = & ab-ac \\ & & & \,(b-c)a & = & ba-ca \\ \end{array}$

• Identity Property $\begin{array}{lccl} \,\mathrm{of addition}\,\mathrm{For any real number}\,a: & & & \,a+0=a \\ \,0\,\mathrm{is the}\,\mathrm{additive identity} & & & \,0+a=a \\ \,\mathrm{of multiplication}\,\mathrm{For any real number}\,a: & & & \,a\cdot{}1=a \\ \,1\,\mathrm{is the}\,\mathrm{multiplicative identity} & & & \,1\cdot{}a=a \\ \end{array}$

• Inverse Property $\begin{array}{lccr} \,\mathrm{of addition}\,\mathrm{For any real number}\,a, & & & \,a+(-a)=0 \\ \,-a\,\mathrm{is the}\,\mathrm{additive inverse}\,\mathrm{of}\,a & & & \\ \,\mathrm{A number and its}\,opposite\,\mathrm{add to zero.} & & & \\ \,\mathrm{of multiplication}\,\mathrm{For any real number}\,a,a\ne{}0 & & & \,a\cdot\frac{1}{a}=1 \\ \,\frac{1}{a}\,\mathrm{is the}\,\mathrm{multiplicative inverse}\,\mathrm{of}\,a & & & \\ \,\mathrm{A number and its}\,reciprocal\,\mathrm{multiply to one.} & & & \\ \end{array}$

• Properties of Zero $\begin{array}{lccl} \,\mathrm{For any real number}\,a, & & & \,a\cdot{}0=0 \\ & & & \,0\cdot{}a=0 \\ \,\mathrm{For any real number}\,a,a\ne{}0, & & & \,\frac{0}{a}=0 \\ \,\mathrm{For any real number}\,a, & & & \,\frac{a}{0}\,\mathrm{is undefined} \\ \end{array}$

##

Review Exercises

Identify Multiples and Factors

In the following exercises, simplify each expression.

Evaluate an Expression

In the following exercises, evaluate the following expressions.

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the following expressions by combining like terms.

Translate an English Phrase to an Algebraic Expression

In the following exercises, translate the phrases into algebraic expressions.

Simplify Expressions with Absolute Value

In the following exercise, fill in $<,>,$ or $=$ for each of the following pairs of numbers.

In the following exercises, simplify.

Add and Subtract Integers

In the following exercises, simplify each expression.

Multiply and Divide Integers

In the following exercise, multiply or divide.

Simplify and Evaluate Expressions with Integers

In the following exercises, simplify each expression.

For the following exercises, evaluate each expression.

Translate English Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

Use Integers in Applications

In the following exercise, solve.

Simplify Fractions

In the following exercises, simplify.

Multiply and Divide Fractions

In the following exercises, perform the indicated operation.

Add and Subtract Fractions

In the following exercises, perform the indicated operation.

Use the Order of Operations to Simplify Fractions

In the following exercises, simplify.

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

Round Decimals

Add and Subtract Decimals

In the following exercises, perform the indicated operation.

Multiply and Divide Decimals

In the following exercises, perform the indicated operation.

Convert Decimals, Fractions and Percents

In the following exercises, convert each decimal to a fraction.

In the following exercises, convert each fraction to a decimal.

In the following exercises, convert each decimal to a percent.

Simplify Expressions with Square Roots

In the following exercises, simplify.

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

In the following exercise, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers for each set of numbers

Locate Fractions and Decimals on the Number Line

In the following exercises, locate the numbers on a number line.

Use the Commutative and Associative Properties

In the following exercises, simplify.

Use the Properties of Identity, Inverse and Zero

In the following exercises, simplify.

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the Distributive Property.

##

In the following exercises, simplify each expression.

1. Use the divisibility tests to determine whether 180 is divisible by 2, by 3, by 5, by 6, and by 10.

2. $14{x}^{2}-9x+11-8{x}^{2}+8x-6$

3. ⓐ eleven times the difference of $y$ and twoⓑ the difference of eleven times $y$ and two

1. ⓐ $-|7|\mathrm{\_\_\_}-|-7|$ⓑ $-8\mathrm{\_\_\_}-|-8|$ⓒ $|-13|\mathrm{\_\_\_}-13$ⓓ $|-12|\mathrm{\_\_\_}-(-12)$

2. $12-3|1-4(4-2)|$

3. ⓐ $15-7$ⓑ $-15-(-7)$ⓒ $-15-7$ⓓ $15-(-7)$

1. ⓐ $23-(-17)$ ⓑ $23+17$

2. ⓐ $-27\mathrm{÷}9$ ⓑ $120\mathrm{÷}(-8)$ ⓒ $4(-14)$ ⓓ $-1(-17)$

3. $6-3|4(1-2)-(7-5)|$

1. the sum of $-4$ and $-9,$ increased by 23

2. Temperature On July 10, the high temperature in Phoenix, Arizona, was 109°, and the high temperature in Juneau, Alaska, was 63°. What was the difference between the temperature in Phoenix and the temperature in Juneau?

3. $-\frac{270{x}^{3}}{198{y}^{2}}$

1. $\frac{6x}{25}\mathrm{÷}\frac{9y}{20}$

2. $\frac{5}{18}+\frac{7}{12}$

3. ⓐ $\frac{5}{8}+\frac{3}{4}$ ⓑ $\frac{5}{8}\mathrm{÷}\frac{3}{4}$

1. $\frac{4\cdot{}3-2\cdot{}5}{-6\cdot{}3+2\cdot{}3}$

2. $\frac{a+b}{a-b}$ when$a=-4,$$b=6$

3. $-23.67+29.84$

1. $79.38-(-17.598)$

2. $(0.563)(100)$

3. $1.84\mathrm{÷}(-0.8)$

1. $-9.6$

2. $\frac{14}{11}$

3. $0.0475$

1. $\sqrt{-121}$

2. $\frac{3}{4},-\frac{3}{4},1\frac{1}{3},-1\frac{2}{3},\frac{7}{2},-\frac{5}{2}$

3. $\frac{5}{8}x+\frac{5}{12}y+\frac{1}{8}x+\frac{7}{12}y$

1. $(\frac{11}{15}+\frac{3}{8})+\frac{5}{8}$

2. $\frac{13}{15}\cdot\frac{9}{17}\cdot\frac{15}{13}$

3. $\frac{5x-7}{0},5x-7\ne{}0$

1. $12(\frac{2}{3}b+\frac{5}{6})$

2. $(x-5)p$

3. $12-6(x+3)$

1. $5(2y+3)-(4y-1)$

2. Combine like terms: $5n+8+2n-1$

3. Translate to an algebraic expression and simplify:

ⓐ eleven less than negative eight

ⓑ the difference of $-8$ and $-3$, increased by 5

1. Round $28.1458$ to the nearest

ⓐ hundredth ⓑ thousandth

2. Locate $\frac{3}{5},2.8,\mathrm{and}-\frac{5}{2}$ on a number line.

3. $16-2|3(1-4)-(8-5)|$

1. $\frac{180}{204}$

2. $\frac{4}{5}\mathrm{÷}(-\frac{12}{25})$

3. $\frac{4(-3+2(3-6))}{3(11-3(2+3))}$

1. $\frac{-\frac{5}{9}}{\frac{10}{21}}$

2. $34.6-100$

3. $-8\mathrm{÷}0.05$

1. $(\frac{8}{13}+\frac{5}{7})+\frac{2}{7}$

2. ⓐ $\frac{0}{9}$ ⓑ $\frac{11}{0}$

3. $6(3y-1)-(5y-3)$