Chapter 3 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.

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Worked example

absolute value

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

integers

Integers are counting numbers, their opposites, and zero ... –3, –2, –1, 0, 1, 2, 3 ...

negative number

A negative number is less than zero.

opposites

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.

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Key Concepts

• Opposite Notation $-a$ means the opposite of the number $a$ The notation $-a$ is read the opposite of $a.$

• Absolute Value Notation The absolute value of a number $n$ is written as $|n|$. $|n|\ge{}0$ for all numbers.

• Addition of Positive and Negative Integers $5+3$$-5+(-3)$ both positive, sum positive both negative, sum negative When the signs are the same, the counters would be all the same color, so add them. $-5+3$$5+(-3)$ different signs, more negatives different signs, more positives Sum negativesum positive When the signs are different, some counters would make neutral pairs; subtract to see how many are left.

• Subtraction of Integers$5-3$$--5-(--3)$$2$$--2$ 2 positives 2 negatives When there would be enough counters of the color to take away, subtract. $--5-3$$5-(--3)$$--8$$8$ 5 negatives, want to subtract 3 positives 5 positives, want to subtract 3 negatives need neutral pairs need neutral pairs When there would not be enough of the counters to take away, add neutral pairs.Table 3.13

• Subtraction Property $a-b=a+(\mathrm{-b})$$a-(\mathrm{-b})=a+b$

• Solve Application Problems Step 1. Identify what you are asked to find. Step 2. Write a phrase that gives the information to find it. Step 3. Translate the phrase to an expression. Step 4. Simplify the expression. Step 5. Answer the question with a complete sentence.

• Multiplication of Signed Numbers To determine the sign of the product of two signed numbers:Same Signs ProductTwo positivesTwo negatives PositivePositiveDifferent Signs ProductPositive • negativeNegative • positive NegativeNegative

• Division of Signed Numbers To determine the sign of the quotient of two signed numbers:Same Signs QuotientTwo positivesTwo negatives PositivePositiveDifferent Signs QuotientPositive • negativeNegative • Positive NegativeNegative

• Multiplication by $-1$ Multiplying a number by $-1$ gives its opposite: $-1a=-a$

• Division by $-1$ Dividing a number by $-1$ gives its opposite: $a\mathrm{÷}(-1)=\mathrm{-a}$

• How to determine whether a number is a solution to an equation. Step 1. Substitute the number for the variable in the equation. Step 2. Simplify the expressions on both sides of the equation. Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

• Properties of Equalities Subtraction Property of Equality Addition Property of Equality$\mathrm{For any numbers}\,a,b,c,$$\mathrm{if}\,a=b\,\mathrm{then}\,a-c=b-c.$$\mathrm{For any numbers}\,a,b,c,$$\mathrm{if}\,a=b\,\mathrm{then}\,a+c=b+c.$

• Division Property of Equality For any numbers $a,b,c,$ and $c\ne{}0$ If $a=b$, then $\frac{a}{c}=\frac{b}{c}$.

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Review Exercises

Locate Positive and Negative Numbers on the Number Line

In the following exercises, locate and label the integer on the number line.

Order Positive and Negative Numbers

In the following exercises, order each of the following pairs of numbers, using $<$ or $\mathrm{>.}$

Find Opposites

In the following exercises, find the opposite of each number.

In the following exercises, simplify.

In the following exercises, evaluate.

Simplify Absolute Values

In the following exercises, simplify.

In the following exercises, evaluate.

In the following exercises, fill in $<,>,\mathrm{or}=$ for each of the following pairs of numbers.

In the following exercises, simplify.

Translate Phrases to Expressions with Integers

In the following exercises, translate each of the following phrases into expressions with positive or negative numbers.

Model Addition of Integers

In the following exercises, model the following to find the sum.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

Add Integers in Applications

In the following exercises, solve.

Model Subtraction of Integers

In the following exercises, model the following.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Translate Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

Subtract Integers in Applications

In the following exercises, solve the given applications.

Multiply Integers

In the following exercises, multiply.

Divide Integers

In the following exercises, divide.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Translate Word Phrases to Algebraic Expressions

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

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

Using the Addition and Subtraction Properties of Equality

In the following exercises, solve.

Model the Division Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters. Then solve it.

Solve Equations Using the Division Property of Equality

In the following exercises, solve each equation using the division property of equality and check the solution.

Translate to an Equation and Solve.

In the following exercises, translate and solve.

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In the following exercises, compare the numbers, using $<\,\mathrm{or}\,>\,\mathrm{or}\,\mathrm{=.}$

In the following exercises, find the opposite of each number.

In the following exercises, simplify.

In the following exercises, evaluate.

In the following exercises, translate each phrase into an algebraic expression and then simplify, if possible.

In the following exercises, solve.

In the following exercises, solve.

In the following exercises, translate and solve.

Key Terms

absolute value

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

integers

Integers are counting numbers, their opposites, and zero ... –3, –2, –1, 0, 1, 2, 3 ...

negative number

A negative number is less than zero.

opposites

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.

##

Key Concepts

• Opposite Notation $-a$ means the opposite of the number $a$ The notation $-a$ is read the opposite of $a.$

• Absolute Value Notation The absolute value of a number $n$ is written as $|n|$. $|n|\ge{}0$ for all numbers.

• Addition of Positive and Negative Integers $5+3$$-5+(-3)$ both positive, sum positive both negative, sum negative When the signs are the same, the counters would be all the same color, so add them. $-5+3$$5+(-3)$ different signs, more negatives different signs, more positives Sum negativesum positive When the signs are different, some counters would make neutral pairs; subtract to see how many are left.

• Subtraction of Integers$5-3$$--5-(--3)$$2$$--2$ 2 positives 2 negatives When there would be enough counters of the color to take away, subtract. $--5-3$$5-(--3)$$--8$$8$ 5 negatives, want to subtract 3 positives 5 positives, want to subtract 3 negatives need neutral pairs need neutral pairs When there would not be enough of the counters to take away, add neutral pairs.Table 3.13

• Subtraction Property $a-b=a+(\mathrm{-b})$$a-(\mathrm{-b})=a+b$

• Solve Application Problems Step 1. Identify what you are asked to find. Step 2. Write a phrase that gives the information to find it. Step 3. Translate the phrase to an expression. Step 4. Simplify the expression. Step 5. Answer the question with a complete sentence.

• Multiplication of Signed Numbers To determine the sign of the product of two signed numbers:Same Signs ProductTwo positivesTwo negatives PositivePositiveDifferent Signs ProductPositive • negativeNegative • positive NegativeNegative

• Division of Signed Numbers To determine the sign of the quotient of two signed numbers:Same Signs QuotientTwo positivesTwo negatives PositivePositiveDifferent Signs QuotientPositive • negativeNegative • Positive NegativeNegative

• Multiplication by $-1$ Multiplying a number by $-1$ gives its opposite: $-1a=-a$

• Division by $-1$ Dividing a number by $-1$ gives its opposite: $a\mathrm{÷}(-1)=\mathrm{-a}$

• How to determine whether a number is a solution to an equation. Step 1. Substitute the number for the variable in the equation. Step 2. Simplify the expressions on both sides of the equation. Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

• Properties of Equalities Subtraction Property of Equality Addition Property of Equality$\mathrm{For any numbers}\,a,b,c,$$\mathrm{if}\,a=b\,\mathrm{then}\,a-c=b-c.$$\mathrm{For any numbers}\,a,b,c,$$\mathrm{if}\,a=b\,\mathrm{then}\,a+c=b+c.$

• Division Property of Equality For any numbers $a,b,c,$ and $c\ne{}0$ If $a=b$, then $\frac{a}{c}=\frac{b}{c}$.

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Review Exercises

Locate Positive and Negative Numbers on the Number Line

In the following exercises, locate and label the integer on the number line.

Order Positive and Negative Numbers

In the following exercises, order each of the following pairs of numbers, using $<$ or $\mathrm{>.}$

Find Opposites

In the following exercises, find the opposite of each number.

In the following exercises, simplify.

In the following exercises, evaluate.

Simplify Absolute Values

In the following exercises, simplify.

In the following exercises, evaluate.

In the following exercises, fill in $<,>,\mathrm{or}=$ for each of the following pairs of numbers.

In the following exercises, simplify.

Translate Phrases to Expressions with Integers

In the following exercises, translate each of the following phrases into expressions with positive or negative numbers.

Model Addition of Integers

In the following exercises, model the following to find the sum.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

Add Integers in Applications

In the following exercises, solve.

Model Subtraction of Integers

In the following exercises, model the following.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Translate Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

Subtract Integers in Applications

In the following exercises, solve the given applications.

Multiply Integers

In the following exercises, multiply.

Divide Integers

In the following exercises, divide.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Translate Word Phrases to Algebraic Expressions

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

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

Using the Addition and Subtraction Properties of Equality

In the following exercises, solve.

Model the Division Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters. Then solve it.

Solve Equations Using the Division Property of Equality

In the following exercises, solve each equation using the division property of equality and check the solution.

Translate to an Equation and Solve.

In the following exercises, translate and solve.

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In the following exercises, compare the numbers, using $<\,\mathrm{or}\,>\,\mathrm{or}\,\mathrm{=.}$

In the following exercises, find the opposite of each number.

In the following exercises, simplify.

In the following exercises, evaluate.

In the following exercises, translate each phrase into an algebraic expression and then simplify, if possible.

In the following exercises, solve.

In the following exercises, solve.

In the following exercises, translate and solve.

1. $4\mathrm{\_\_}8$

2. $-5\mathrm{\_\_}-10$

3. $2\mathrm{\_\_}-7$

4. $6$

1. 1. ⓐ $\,-(8)\,$

2. ⓑ $\,-(-8)$

2. $-x,\,\mathrm{when}\,$

1. ⓐ $\,x=32\,$

2. ⓑ $\,x=-32$

3. $-|-y|\,\mathrm{when}\,y=33$

4. $-|-4|\mathrm{\_\_}4$

1. $-|-6|\mathrm{\_\_}-6$

2. $-(-55)\,\mathrm{and}\,-|-55|$

3. $|9-3|-|5-12|$

4. the opposite of $16$

1. negative $3$

2. a temperature of $10$ below zero

3. $-75+25$

4. $11+(-15)+3$

1. $n+4\,\mathrm{when}\,$

1. ⓐ $\,n=-1\,$

2. ⓑ $\,n=-20$

2. ${(x+y)}^{3}\,\mathrm{when}\,x=-4,y=1$

3. $\mathrm{the sum of -8 and 2}$

4. $\mathrm{10 more than the sum of -5 and -6}$

1. Temperature On Monday, the high temperature in Denver was $\mathrm{-4 degrees.}$ Tuesday’s high temperature was $\mathrm{20 degrees}$ more. What was the high temperature on Tuesday?

2. $-31-7$

3. $-52-(-17)-23$

4. $(1-7)-(3-8)$

1. $x-7\,\mathrm{when}\,$

1. ⓐ $\,x=5\,$

2. ⓑ $\,x=-4$

2. the difference of $-12\,\mathrm{and}\,5$

3. Temperature One morning the temperature in Bangor, Maine was $\mathrm{18 degrees.}$ By afternoon, it had dropped $\mathrm{20 degrees.}$ What was the afternoon temperature?

4. $-9\cdot{}4$

1. $56\mathrm{÷}(-8)$

2. $-96\mathrm{÷}12$

3. $45\mathrm{÷}(-1)$

4. $5(-9)-3(-12)$

1. $-{3}^{4}$

2. $7x-3\,\mathrm{when}\,x=-9$

3. $5a+8b\,\mathrm{when}\,a=-2,b=-6$

4. the product of $-12$ and $6$

1. $5x-10=-35$

1. ⓐ $\,x=-9$

2. ⓑ $\,x=-5$

3. ⓒ $\,x=5$

2. $a+14=2$

3. $c+(-10)=-17$

4. $\mathrm{The product of -6 and}\,y\,\mathrm{is}\,-42$

1. Four more than $m$ is $-48.$

2. Describe how you have used two topics from this chapter in your life outside of your math class during the past month.

3. 1. ⓐ $\,-6\mathrm{\_\_}3\,$

2. ⓑ $\,-1\mathrm{\_\_}-4$

4. 1. ⓐ $\,-7\,$

2. ⓑ $\,8$

1. $-15+(-12)$

2. ${(-2)}^{3}$

3. the difference of −7 and −4

4. Early one morning, the temperature in Syracuse was $-8\mathrm{^{\circ}F.}$ By noon, it had risen $\mathrm{12^{\circ}.}$ What was the temperature at noon?

1. $n+6=5$

2. $-9r=-54$

3. $\mathrm{Eight less than}\,y\,\mathrm{is -32.}$