Download Chapter 4 Study Guide

Chapter 4 Study Guide
Lesson 4.1: The Distributive Property
Example 1
Evaluate Numeric Expressions
Use the Distributive Property to write each expression as an equivalent expression.
Then evaluate the expression.
a. 3(5 – 2)
3(5 – 2) = 3  5 – 3  2
= 15 – 6
=9
Multiply.
Subtract.
b. (4 + 6)2
(4 + 6)2 = 4  2 + 6  2
= 8 + 12
= 20
Real-World Example 2
Multiply.
Add.
Use the Distributive Property
UNIFORMS The uniform for a fast-food restaurant job costs $39. Find the total cost of
uniforms for 7 new employees.
Understand You know how many uniforms are needed and how each uniform costs.
You need to find the total cost of the uniforms.
Plan
You can use the Distributive Property and mental math to find the total
cost of the uniforms. To find the total cost mentally, find 7($40 – $1).
7($40 – $1) = 7($40) − 7($1)
Solve
Distributive Property
= $280 − $7
= $273
Multiply.
Subtract.
The total cost is $273.
Check
You can check your result by multiplying 7  $40 or $280. The answer
seems reasonable. 
Example 3
Simplify Algebraic Expressions
Use the Distributive Property to write each expression as an equivalent algebraic
expression.
a. 4(3 + m)
4(3 + m) = 4  3 + 4m
= 12 + 4m
Simplify.
b. (p + 2)5
(p + 2)5 = p  5 + 2  5
= 5p + 10
Example 4
Simplify.
Simplify Expressions with Subtraction
Use the Distributive Property to write each expression as an equivalent algebraic
expression.
** If you have a subtraction sign,
change it to addition and change the second number to its opposite. **
a. 5(w – 7)
5(w – 7) = 5[w + (-7)]
= 5  w + 5  (-7)
= 5w + (-35)
= 5w – 35
Rewrite w – 7 as w + (-7).
Distributive Property
Simplify.
Definition of subtraction
b. –2(y – 4)
-2(y – 4) = -2[y + (-4)]
= -2  y + (-2)(-4)
= -2y + 8
Rewrite y – 4 as y + (-4).
Distributive Property
Simplify.
Lesson 4.1 ½ : Factoring Linear Expressions
A monomial is a number, a variable, or a product of a number and one or more variables.
Monomials
25, x, 40x
Not Monomials
x + 4, 40x+120
To factor a number means to write it as a product of its factors. A monomial can be
factored using the same method that you would use to factor a number. The greatest
common factor (GCF) of two monomials is the greatest monomial that is a factor of both.
Example 1: Find the GCF of Monomials
a.) Find the GCF of 4x and 12x
Think, what are the factors?:
4x = 1, 2, 4, x
Write the factors of 4x and 12x
12x = 1, 2 , 3, 4, 6, 12, x
4x and 12x have factors of 1, 4, x
The GCF of 4x and 12x is 4x.
b.) Find the GCF of 18a and 20ab
Think, what are the factors?:
18a = 1, 2, 9, 18, a
Write the factors of 18a and 20ab
20ab = 1, 2, 4, 5, 20, a, b
18a and 20ab have 1, 2, a in common.
The GCF of 18a and 20ab is 2a.
You use the Distributive Property and the work backward strategy to express an algebraic
expression as a product of its factors. An algebraic expression is in factored form when it
is expressed as the product of its factors.
8x + 4 = 4(2x) + 4(1)
4(2x + 1)
Example 2: Factor Algebraic Expressions
a.) 3x + 9
Think, what are the factors?:
Find the GCF: 3x = 1, 3, x
3x and 9 have 1, 3 in common.
9 = 1, 3, 9
The GCF of 3x and 9 is 3.
3x + 9 = 3(x) + 3(3)
3(x+3)
b.) 12x + 7
Think, what are the factors?:
Find the GCF: 12x= 1, 2, 3, 4, 6, 12, x
12x and 7 have 1 in common.
7 = 1, 7
Since the only common factor is 1, 12x +7 cannot be factored.
c.) 4x + 28
Think, what are the factors?:
Find the GCF: 4x = 1, 2, 4, x
28 = 1, 2, 4, 7, 14
The GCF of 4x and 28 is 4.
4x + 28 = 4(x) + 4(7)
4(x + 7)
4x and 28 have 1, 2, 4 in common.
Lesson 4.2: Simplifying Expressions
Example 1
Identify Like Terms
Identify the like terms in the following expressions.
a. 7y + 1 + 8y
7y and 8y are like terms since the variables are the same.
b. 9a + 2 + 6 + 4a
9a and 4a are like terms since the variables are the same. Constant terms 2 and 6 are
also like terms.
Example 2
Identify Parts of an Expression
Identify the terms, like terms, coefficients, and constants in the expression
x + 5y – 2y – 3
x + 5y – 2y – 3 = x + 5y + (-2y) + (-3)
= 1x + 5y + (-2y) + (-3)
Definition of subtraction
Identity Property
The terms are x, 5y, -2y, and -3. The like terms are 5y and -2y. The coefficients are 1, 5,
and -2. The constant is -3.
Example 3
Simplify Algebraic Expressions
Simplify each expression.
a. 3x + 9 + 4x
3x + 9 + 4x = 3x + 4x + 9
= (3 + 4)x + 9
= 7x + 9
Commutative Property
Distributive Property
Simplify.
b. 8 – 5m + m – 3
8 – 5m + m – 3 = 8 + (-5m) + m + (-3)
= 8 + (-5m) + (1m) + (-3)
= (-5m) + (1m) + 8 + (-3)
= (-5 + 1)m + 8 + (-3)
= -4m + 5
Definition of subtraction
Identity Property
Commutative Property
Distributive Property
Simplify.
c. a + 5(2a + 3b)
a + 5(2a + 3b) = a + 5(2a) + 5(3b)
= a + 10a + 15b
= 1a + 10a + 15b
= (1 + 10)a + 15b
= 11a + 15b
Real-World Example 4
Distributive Property
Simplify.
Identity Property
Distributive Property
Simplify.
Write and Simplify Algebraic Expressions
AGES Emily and Kate are sisters. Emily is four years younger than Kate. Write
an expression in simplest form that represents the sum of Emily and Kate’s ages.
Words
Emily’s age plus
Variables
Let x = Kate’s age.
Kate’s age
Let x – 4 = Emily’s age.
Expression
(x – 4) + x = x + (x – 4)
= (x + x) – 4
= (1x + 1x) – 4
= (1 + 1)x – 4
= 2x – 4
(x – 4) + x
Commutative Property
Associative Property
Identity Property
Distributive Property
Simplify.
The expression 2x – 4 represents the sum of Emily and Kate’s ages.
Lesson 4.3: Solving Equations by Addition or Subtraction
Example 1
Solve Equations by Adding
Solve each equation. Check your solution and graph it on a number line.
a. –3 + x = 18
–3 + x = 18
+3
=+3
x = 21
Write the equation.
Addition Property of Equality
To check that 21 is the solution, replace x with 21 in the original equation.
Check
–3 + x = 18
–3 + 21 ≟ 18
18 = 18 
Write the equation.
Check to see whether this sentence is true.
The sentence is true.
The solution is 21. To graph 21, draw a dot at 21 on a number line.
13
17 18
–5 14
–4 15
–3 16
–2 –1
0 19
1 20
2
21
3 22
4 23
5
b. 8.2 = q – 1.5
8.2 = q – 1.5
Write the equation.
8.2= q + -1.5
+ 1.5 = + 1.5
9.7 = q
Change the subtraction to “add the opposite”
Addition Property of Equality
The solution is 9.7.
Check your solution.
To graph 9.7, draw a dot at 9.7 on a number line.
9.7

1
2
3
4
5
6
7
8
9
10
12
Example 2
Solving Equations by Subtracting
Solve each equation. Check your solution.
a. 9 + b = 4
9+b= 4
–9
=–9
b = –5
Write the equation.
Subtraction Property of Equality
Additive Inverse and Identity Properties
To check that –5 is the solution, replace b with –5 in the original equation.
Check
9+b=4
Write the equation.
?
9 + (-5)  4
4=4 
Check to see whether this sentence is true.
The sentence is true.
b. 2.8 + y = 0.4
2.8 + y = 0.4
– 2.8
= – 2.8
y = –2.4
Write the equation.
Subtraction Property of Equality
Additive Inverse and Identity Properties
The solution is -2.4. Check your solution.
Real-World Example 3
Solve by Subtracting
BASEBALL Carlos had 12 more hits during baseball season than Peter. Carlos
had 73 hits during the season. Write and solve an equation to find the number of
hits that Peter had.
Words
Carlos’s hits equals Peter’s hits + 12.
Variable
Let x = the number of hits that Peter had.
Equation 73 = x + 12
73 = x + 12
– 12= – 12
61 = x
Peter had 61 hits.
Write the equation.
Subtraction Property of Equality.
Additive Inverse and Identity Properties
Lesson 4.4: Solving Equations by Multiplying or Dividing
Lesson 4.5: Solving Two Step Equations
Example 3
Solve a Two-Step Equation
y
Solve 8 +
= 14. Check your solution.
3
y
8+
= 14
Write the equation.
3
-8
= –8
Subtraction Property of Equality
y
=6
Simplify.
3
y
3
=36
Multiplication Property of Equality
3
y = 18
Simplify. Check your solution.
Example 4
Equations with Negative Coefficients
Solve 6 = 9 – x. Check your solution.
6=9–x
Write the equation.
6 = 9 – 1x
Identity Property; x = 1x
6 = 9 + (-1x)
Definition of subtraction
-9 + 6 = -9 + 9 + (-1x)
Addition Property of Equality
-3 = -1x
Simplify.
3
 1x
=
Division Property of Equality
1
1
3=x
Simplify. Check your solution.
Example 5
Combine Like Terms Before Solving
Solve 3w – 7w – 11 = 25. Check your solution.
3w – 7w – 11 = 25
Write the equation.
-4w – 11 = 25
Distributive Property; 3w – 7w = –4w
-4w +-11 = 25
Change subtraction to “add the opposite”
+11 = +11
Addition Property of Equality
-4w = 36
Simplify.
 4w
36
=
Division Property of Equality
4
4
w = -9
Simplify. Check your solution.
Lesson 4.6: Writing Equations
Example 2
Translate Sentences into Equations
Translate each sentence into an equation.
Sentence
Equation
a. Three less than twice a number is –12.
2n – 3 = -12
b. Mara has 15 ponytail holders, which is 6 more than 3 times
the number that Janelle has.
15 = 3n + 6
c. The quotient of a number and 4, decreased by 2, is 5.
n
4-2=5
Example 3
Write and Solve an Equation
BASKETBALL Olivia scored twenty-four points in a basketball game. This is six
more than three times the number of points that Madison scored. Find the number
of points that Madison scored.
Let n = the number of points that Madison scored. Then, 3n + 6 = the number of points
that Olivia scored.
24 = 3n + 6
–6=
–6
18 = 3n
6=n
Write the equation.
Subtraction Property of Equality
Simplify.
Mentally divide each side by 3.
Madison scored 6 points.
Real-World Example 4
Solve a Two-Step Verbal Problem