Chapter 2 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

coefficient

The constant that multiplies the variable(s) in a term is called the coefficient.

composite number

A composite number is a counting number that is not prime.

divisibility

equation

An equation is made up of two expressions connected by an equal sign.

evaluate

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number.

expressions

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

least common multiple

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

like terms

Terms that are either constants or have the same variables with the same exponents are like terms.

multiple of a number

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 itself.

solution of an equation

A solution to an equation is a value of a variable that makes a true statement when substituted into the equation. The process of finding the solution to an equation is called solving the equation.

term

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

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Key Concepts

• Equality Symbol $a=b$ is read as $a$ is equal to $b$ The symbol $=$ is called the equal sign.

• Inequality $a<b$ is read $a$ is less than $b$$a$ is to the left of $b$ on the number line $a>b$ is read $a$ is greater than $b$$a$ is to the right of $b$ on the number line

Table

2.14

• Exponential Notation For any expression ${a}^{n}$ is a factor multiplied by itself $n$ times, if $n$ is a positive integer. ${a}^{n}$ means multiply $n$ factors of $a$ The expression of ${a}^{n}$ is read $a$ to the $n\mathrm{th}$ power.

Order of Operations When simplifying mathematical expressions perform the operations in the following order:

• Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

• Exponents: Simplify all expressions with exponents.

• Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.

• Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.

• Combine like terms. Step 1. Identify like terms. Step 2. Rearrange the expression so like terms are together. Step 3. Add the coefficients of the like terms

• 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.

• Subtraction Property of Equality For any numbers $a$, $b$, and $c$,if $a=b$ then $a-c=b-c$

• Solve an equation using the Subtraction Property of Equality. Step 1. Use the Subtraction Property of Equality to isolate the variable. Step 2. Simplify the expressions on both sides of the equation. Step 3. Check the solution.

• Addition Property of Equality For any numbers $a$, $b$, and $c$,if $a=b$ then $a+c=b+c$

• Solve an equation using the Addition Property of Equality. Step 1. Use the Addition Property of Equality to isolate the variable. Step 2. Simplify the expressions on both sides of the equation. Step 3. Check the solution.

• Factors If $a\cdot{}b=m$, then $a$ and $b$ are factors of $m$, and $m$ is the product of $a$ and $b$.

• Find all the factors of a counting number. Step 1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor. If the quotient is a counting number, the divisor and quotient are a pair of factors. If the quotient is not a counting number, the divisor is not a factor. Step 2. List all the factor pairs. Step 3. Write all the factors in order from smallest to largest.

• Determine if a number is prime. Step 1. Test each of the primes, in order, to see if it is a factor of the number. Step 2. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found. Step 3. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.

• Find the prime factorization of a composite number using the tree method. Step 1. Find any factor pair of 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. Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process. Step 4. Write the composite number as the product of all the circled primes.

• Find the prime factorization of a composite number using the ladder method. Step 1. Divide the number by the smallest prime. Step 2. Continue dividing by that prime until it no longer divides evenly. Step 3. Divide by the next prime until it no longer divides evenly. Step 4. Continue until the quotient is a prime. Step 5. Write the composite number as the product of all the primes on the sides and top of the ladder.

• Find the LCM by listing multiples. Step 1. List the first several multiples of each number. Step 2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number. Step 3. Look for the smallest number that is common to both lists. Step 4. This number is the LCM.

• Find the LCM using the prime factors method. Step 1. Find the prime factorization of each number. Step 2. Write each number as a product of primes, matching primes vertically when possible. Step 3. Bring down the primes in each column. Step 4. Multiply the factors to get the LCM.

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Review Exercises

Use Variables and Algebraic Symbols

In the following exercises, translate from algebra to English.

Identify Expressions and Equations

In the following exercises, determine if each is an expression or equation.

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

In the following exercises, write in expanded form.

In the following exercises, simplify each expression.

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

Evaluate an Expression

In the following exercises, evaluate the following expressions.

Identify Terms, Coefficients and Like Terms

In the following exercises, identify the terms in each expression.

In the following exercises, identify the coefficient of each term.

In the following exercises, identify the like terms.

Simplify Expressions by Combining Like Terms

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

Translate English Phrases to Algebraic Expressions

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

Determine Whether a Number is a Solution of an Equation

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

Model the Subtraction Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality.

Solve Equations using the Subtraction Property of Equality

In the following exercises, solve each equation using the subtraction property of equality.

Solve Equations using the Addition Property of Equality

In the following exercises, solve each equation using the addition property of equality.

Translate English Sentences to Algebraic Equations

In the following exercises, translate each English sentence into an algebraic equation.

Translate to an Equation and Solve

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

Mixed Practice

In the following exercises, solve each equation.

Identify Multiples of Numbers

In the following exercises, list all the multiples less than $50$ for each of the following.

Use Common Divisibility Tests

In the following exercises, using the divisibility tests, determine whether each number is divisible by $2,\mathrm{by}\,3,\mathrm{by}\,5,\mathrm{by}\,6,\mathrm{and by}\,10.$

Find All the Factors of a Number

In the following exercises, find all the factors of each number.

Identify Prime and Composite Numbers

In the following exercises, identify each number as prime or composite.

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number.

Find the Least Common Multiple of Two Numbers

In the following exercises, find the least common multiple of each pair of numbers.

##

In the following exercises, translate from an algebraic equation to English phrases.

In the following exercises, identify each as an expression or equation.

In the following exercises, simplify, using the order of operations.

In the following exercises, evaluate each expression.

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

In the following exercises, solve each equation.

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

Key Terms

coefficient

The constant that multiplies the variable(s) in a term is called the coefficient.

composite number

A composite number is a counting number that is not prime.

divisibility

equation

An equation is made up of two expressions connected by an equal sign.

evaluate

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number.

expressions

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

least common multiple

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

like terms

Terms that are either constants or have the same variables with the same exponents are like terms.

multiple of a number

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 itself.

solution of an equation

A solution to an equation is a value of a variable that makes a true statement when substituted into the equation. The process of finding the solution to an equation is called solving the equation.

term

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

##

Key Concepts

• Equality Symbol $a=b$ is read as $a$ is equal to $b$ The symbol $=$ is called the equal sign.

• Inequality $a<b$ is read $a$ is less than $b$$a$ is to the left of $b$ on the number line $a>b$ is read $a$ is greater than $b$$a$ is to the right of $b$ on the number line

Table

2.14

• Exponential Notation For any expression ${a}^{n}$ is a factor multiplied by itself $n$ times, if $n$ is a positive integer. ${a}^{n}$ means multiply $n$ factors of $a$ The expression of ${a}^{n}$ is read $a$ to the $n\mathrm{th}$ power.

Order of Operations When simplifying mathematical expressions perform the operations in the following order:

• Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

• Exponents: Simplify all expressions with exponents.

• Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.

• Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.

• Combine like terms. Step 1. Identify like terms. Step 2. Rearrange the expression so like terms are together. Step 3. Add the coefficients of the like terms

• 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.

• Subtraction Property of Equality For any numbers $a$, $b$, and $c$,if $a=b$ then $a-c=b-c$

• Solve an equation using the Subtraction Property of Equality. Step 1. Use the Subtraction Property of Equality to isolate the variable. Step 2. Simplify the expressions on both sides of the equation. Step 3. Check the solution.

• Addition Property of Equality For any numbers $a$, $b$, and $c$,if $a=b$ then $a+c=b+c$

• Solve an equation using the Addition Property of Equality. Step 1. Use the Addition Property of Equality to isolate the variable. Step 2. Simplify the expressions on both sides of the equation. Step 3. Check the solution.

• Factors If $a\cdot{}b=m$, then $a$ and $b$ are factors of $m$, and $m$ is the product of $a$ and $b$.

• Find all the factors of a counting number. Step 1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor. If the quotient is a counting number, the divisor and quotient are a pair of factors. If the quotient is not a counting number, the divisor is not a factor. Step 2. List all the factor pairs. Step 3. Write all the factors in order from smallest to largest.

• Determine if a number is prime. Step 1. Test each of the primes, in order, to see if it is a factor of the number. Step 2. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found. Step 3. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.

• Find the prime factorization of a composite number using the tree method. Step 1. Find any factor pair of 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. Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process. Step 4. Write the composite number as the product of all the circled primes.

• Find the prime factorization of a composite number using the ladder method. Step 1. Divide the number by the smallest prime. Step 2. Continue dividing by that prime until it no longer divides evenly. Step 3. Divide by the next prime until it no longer divides evenly. Step 4. Continue until the quotient is a prime. Step 5. Write the composite number as the product of all the primes on the sides and top of the ladder.

• Find the LCM by listing multiples. Step 1. List the first several multiples of each number. Step 2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number. Step 3. Look for the smallest number that is common to both lists. Step 4. This number is the LCM.

• Find the LCM using the prime factors method. Step 1. Find the prime factorization of each number. Step 2. Write each number as a product of primes, matching primes vertically when possible. Step 3. Bring down the primes in each column. Step 4. Multiply the factors to get the LCM.

##

Review Exercises

Use Variables and Algebraic Symbols

In the following exercises, translate from algebra to English.

Identify Expressions and Equations

In the following exercises, determine if each is an expression or equation.

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

In the following exercises, write in expanded form.

In the following exercises, simplify each expression.

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

Evaluate an Expression

In the following exercises, evaluate the following expressions.

Identify Terms, Coefficients and Like Terms

In the following exercises, identify the terms in each expression.

In the following exercises, identify the coefficient of each term.

In the following exercises, identify the like terms.

Simplify Expressions by Combining Like Terms

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

Translate English Phrases to Algebraic Expressions

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

Determine Whether a Number is a Solution of an Equation

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

Model the Subtraction Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality.

Solve Equations using the Subtraction Property of Equality

In the following exercises, solve each equation using the subtraction property of equality.

Solve Equations using the Addition Property of Equality

In the following exercises, solve each equation using the addition property of equality.

Translate English Sentences to Algebraic Equations

In the following exercises, translate each English sentence into an algebraic equation.

Translate to an Equation and Solve

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

Mixed Practice

In the following exercises, solve each equation.

Identify Multiples of Numbers

In the following exercises, list all the multiples less than $50$ for each of the following.

Use Common Divisibility Tests

In the following exercises, using the divisibility tests, determine whether each number is divisible by $2,\mathrm{by}\,3,\mathrm{by}\,5,\mathrm{by}\,6,\mathrm{and by}\,10.$

Find All the Factors of a Number

In the following exercises, find all the factors of each number.

Identify Prime and Composite Numbers

In the following exercises, identify each number as prime or composite.

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number.

Find the Least Common Multiple of Two Numbers

In the following exercises, find the least common multiple of each pair of numbers.

##

In the following exercises, translate from an algebraic equation to English phrases.

In the following exercises, identify each as an expression or equation.

In the following exercises, simplify, using the order of operations.

In the following exercises, evaluate each expression.

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

In the following exercises, solve each equation.

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

1. $3\cdot{}8$

2. $24\mathrm{÷}6$

3. $50\ge{}47$

4. $n+4=13$

1. $5+u=84$

2. $4y-11$

3. $2\cdot{}2\cdot{}2$

4. $x\cdot{}x\cdot{}x\cdot{}x\cdot{}x\cdot{}x$

1. ${8}^{4}$

2. ${y}^{5}$

3. $12{n}^{2}+3n+1$

4. $5{x}^{2},3,5{y}^{2},3x,x,4$

1. $15a+9a$

2. $4x+7x+3x$

3. $8n+2+4n+9$

4. $7{y}^{2}+2y+11+3{y}^{2}-8$

1. the difference of $x$ and $6$

2. the product of $3n$ and $9$

3. $5$ times the sum of $y$ and $1$

4. Jack bought a sandwich and a coffee. The cost of the sandwich was $\text{\textdollar 3}$ more than the cost of the coffee. Call the cost of the coffee $c.$ Write an expression for the cost of the sandwich.

1. $y+16=40$

1. ⓐ $24$

2. ⓑ $56$

2. $4n+12=36$

1. ⓐ $6$

2. ⓑ $12$

3. $15x-5=10x+45$

1. ⓐ $2$

2. ⓑ $10$

4. $c+8=14$

1. $23=x+12$

2. $y-7=16$

3. $19=p-15$

4. The sum of $7$ and $33$ is equal to $40.$

1. The product of $4$ and $8$ is equal to $32.$

2. Twice the difference of $n$ and $3$ gives $76.$

3. Eight more than $x$ is equal to $35.$

4. The difference of $q$ and $18$ is $57.$

1. $h-15=27$

2. $z+52=85$

3. $27=q+19$

4. $31=v-25$

1. $3$

2. $8$

3. $96$

4. $420$

1. $30$

2. $180$

3. $19$

4. $121$

1. $84$

2. $350$

3. Describe how you have used two topics from The Language of Algebra chapter in your life outside of your math class during the past month.

4. $15-x$

1. $x+6=9$

2. 1. ⓐ Write $n\cdot{}n\cdot{}n\cdot{}n\cdot{}n\cdot{}n$ in exponential form.

2. ⓑ Write ${3}^{5}$ in expanded form and then simplify.

3. $5$ more than $x$

4. three times the difference of $a\,\mathrm{and}\,b$

1. $n-6=25$

2. $15$ less than $y$ is $32.$

3. List all the multiples of $4$ that are less than $50.$

4. Find the prime factorization of $1080.$