DEV 108
Supplement
Inquiry Sections
DEV 108 Notes
Revised Winter 2007
p. 1
Inequalities
Activity 8.1
Citizens of the United States must be at least 18 years old to vote.
Is there a maximum age restriction for voting?
People over 12 years old are not allowed to participate in the annual Spring Fun Day.
Is there a minimum age requirement?
Can you describe other situations where there may be an beginning point but no end point?
In math we can solve for a beginning point for an infinity of answers. We can solve these
problems as inequalities.
Inequalities are two expressions separated by one of four inequality signs:
< … Less than
> … Greater than
< … Less than or equal to
> … Greater than or equal to
When solving inequalities, your goal is to get the variable by itself. To do this you perform
the inverse operation (opposite operation). You are “undoing” what was done.
Solving inequalities is done the same way you solve equations.
Steps for solving equalities:
Step 1: Distribute any multiplication
Step 2: Combine any like terms
Step 3: “Undo” any addition or subtraction
Step 4: “Undo” any multiplication or division
The only catch in the process is that the inequality sign changes direction when you multiply or
divide by negative number to get the variable by itself…
DEV 108 Notes
Revised Winter 2007
p. 2
Why do you think the inequality sign needs to be reversed when dividing or multiplying by
a negative number?
Practice with Solving and Graphing Inequalities
Directions: Practice: Solve each inequality.
a) 14 - 6x > 3( x -2)
b) -5 + ½ y > 6
c) 10 + 5y < 25
d) 6 - 3x > 12
DEV 108 Notes
Revised Winter 2007
p. 3
Inequality Translations use the Inequality Symbols:
 Inequalities are comparisons that are not equal
 A variable will be used to represent the “unknown”
 The symbols used with inequalities are: < , > , < , or >
 The following phrases are some clues to inequalities:
at least
o more than
greater than
no greater
greater than or equal to
less than or equal to
smaller than
minimum
Examples:
There must be a minimum of 12 people…
Bailey weighs more than 35 pounds…
The box can hold no more than 52 parts…
at most
less
less than
maximum
x > 12 people
x > 35 pounds
x < 52 parts
Practice: Write an inequality expression for each.
a) A child must be at least 65 pounds to ride without a seatbelt.
b) The weight limit on the elevator is 2,000 pounds.
c) Bill must hit more than 3 homeruns to beat his previous record.
The same process used for translating equation word problems can be used for solving
inequalities. Just don’t forget to look for inequality clue word and use inequality symbols.
Remember that the inequality sign will change directions when multiplying or dividing by a
negative number to get the variable by itself.
Example 1: John sells medical supplies. He is paid $900 a month plus 5% of all products that
he sells. If he wants to earn more than $4000 a month, what is the minimum amount that he
must sell?
5% of sales + $900 > $4000
0.05x + $900
> $4000
- $900
- $900
0.05x
> $3100
0.05x
0.05
x
Let x = amount of sales
> $3100
0.05
> $62,000
He’d have to sell more than
$62,000 in products
DEV 108 Notes
Revised Winter 2007
p. 4
Example 2: Patti is having her bathroom installed. The labor is $50 an hour. If her budget is
for $6000, and the cost of the materials is $3300, what is the maximum amount of time that
she can afford for the job?
$3300 + ($50/hour)(number of hours of labor) < $6000
$3300 + $50x
- $3300
$50x
< $6000
-$3300
< $2700
Let x = number of hours
of labor
$50x
$50
< $2700
$50
The job would need
to be completed in
< 54 hrs
less than 54 hours of
labor
x
Practice:
a) Dan wants to build a bookcase that is 3 feet wide. It has a top and 1 additional shelf. If
Dan has a total of 13 feet of boards, what is the greatest height that he can build the
bookcase? (Think perimeter)
What is the clue that this is an inequality?
Solve:
b) Alex is making a triangular wind chime. Two of the sides measure 41” and 54” respectively.
If he only has 150” of material to work with, what is the maximum length of the remaining
side?
What is the clue that this is an inequality?
Solve:
DEV 108 Notes
Revised Winter 2007
p. 5
c) Harry is buying gifts for the holiday. He has spent $85, $65, $35, $100, and $49. If his
budget is $500, what is the maximum amount that he can spend on the remaining gifts?
What is the clue that this is an inequality?
Solve:
DEV 108 Notes
Revised Winter 2007
p. 6
Division with Exponents
Activity 8.2
Keeping in mind that division is the inverse of multiplication, what do you think would be
the process for dividing exponents. Remember the rules for multiplying exponents. Think
opposite.
Use the following division problem for your example:
7a2b
14a2b2
The following are the steps for dividing with exponents.
How close was the process that you used?
Step 1: Divide or reduce the numerical coefficients if necessary
Step 2: Determine if the base value goes on top, on the bottom, or cancels out…write it in
the appropriate spot (if the larger exponent for the given
base is on top, that base value
goes on top…if the larger exponent for the
given base is on the bottom, that base value goes
on the bottom…if the exponent is the same on the top and the bottom, the base value cancels
out…)
Step 3: Subtract the exponent values for the like variable(s) and put the
answer next to
the appropriate base value in the answer
Examples:
x4
x2
= x2
2a2
4a5
12x4y5z = 3x2
16x2y7z 4y2
Practice:
a)
x5
x3
=
1
2a3
-35abc = -5
7abc
7x4y9 = 7y6
12x7y3
12x3
b)
p5
p9
c)
7a2b
14a2b2
d)
a2b3c4
a2b3c4
DEV 108 Notes
Revised Winter 2007
p. 7
e)
12a9b4
16a7b7
i)
24a11b2
8a7b11
f)
j)
25x6y8 g)
-5x3y3
11x7y7
-9x3y3
12abc
18a2b
k)
h)
7ab9c
11a2b
l)
4yz
12xyz
34xyz
17xyz
DEV 108 Notes
Revised Winter 2007
p. 8
Trinomial Factoring
Review of Factoring:
o Factoring expressions is “undoing” the distributive property.
o Factoring indicates that you are able to divide out a common value from each term.
o EXAMPLES:
(6x + 10)
(8x – 12)
2(3x + 5)
4(2x – 3)
Factoring Polynomials into two Binomials:
Factoring is working backwards from “FOILing” or undoing the multiplication of two
binomials.
Activity 8.3
Earlier in the course you multiplied binomials by binomials.
For example: ( x + 2 )( x + 4 ) = x + 6x + 8 The multiplication resulted in a trinomial. The binomials
being multiplied were the factors of the resulting trinomial.
Trinomial factoring is reversing the process of binomial multiplication.
2
Take the following product of binomial multiplication and find its two binomial factors:
x - 10x + 21
2
What process did you use?
Methods for Factoring Trinomials Into Two Binomials
Think back to the patterns of multiplying two binomials:
1) (x + 2)(x + 3) The second terms are positive. This will result in all of the terms in the
trinomial product being positive: x + 5x + 6
2
2) (x – 2)(x – 3) The second terms are negative. This will result in a negative middle term and
a positive third term in the trinomial product: x -5x + 6
2
3) (x – 2)(x + 3) or (x + 2)(x – 3) The signs of the second terms are mixed. This will result in
A negative third term. The second term will take the sign of the larger number.
DEV 108 Notes
Revised Winter 2007
p. 9
Use the patterns to help with factoring…When factoring a polynomial into two binomials:
Step 1: Identify the pattern and set up your parenthesis
Step 2: Determine the factors that are used (when multiplied together
you have the third term…when combined together you have
the middle
term)
Step 3: Insert factors into the parenthesis
EXAMPLES of factoring a polynomial into two binomials:
Example 1:
x2 + 3x + 2
Identify the pattern and set up your parenthesis…
(x + ) (x + )
Determine the factors that are multiplied together
to get the last term in the original problem and when
combined give you the middle term…
1 and 2
Insert factors into your parenthesis…
(x + 1) (x + 2)
You are finished…
Example 2:
x2 - 4x + 3
Identify the pattern and set up your parenthesis…
(x - ) (x 1 and 3
)
Determine the factors that are multiplied together
to get the last term in the original problem and when
combined give you the middle term…
Insert factors into your parenthesis…
(x - 1) (x - 3)
You are finished…
DEV 108 Notes
Revised Winter 2007
p. 10
Example 3:
x2 + 2x – 15
(x + ) (x - )
3 and 5
(x + 5) (x - 3)
Identify the pattern and set up your parenthesis…
Determine the factors that are multiplied together
to get the last term in the original problem and when
combined give you the middle term…the larger
number goes in the parenthesis with the addition sign
since the first operation sign is an addition sign…
Insert factors into your parenthesis…
You are finished…
Example 4:
x2 - x - 12
Identify the pattern and set up your parenthesis…
(x + ) (x - )
3 and 4
(x + 3) (x - 4)
Determine the factors that are multiplied together
to get the last term in the original problem and when
combined give you the middle term…the larger
number goes in the parenthesis with the subtraction
sign since the first operation sign is a subtraction
sign…
Insert factors into your parenthesis…
You are finished…
Example 5:
x2 - 25
Identify the pattern and set up your parenthesis…
(x + ) (x - )
If there are only two terms, the second term will be
a perfect square…the factors will be the square root
of that number…
5 and 5
Insert factors into your parenthesis…
(x + 5) (x - 5)
You are finished…
DEV 108 Notes
Revised Winter 2007
p. 11
Example 6:
x2 - 14x + 49
Identify the pattern and set up your parenthesis…
(x - ) (x - )
Notice that the third term is a perfect square…check
to see if the middle term is the numerical coefficient
of the first term and the square root of the third
term doubled…identify the square root of the third
term and insert it into the parenthesis…
7
(x - 7) (x - 7)
Insert factors into your parenthesis…
You are finished…
Practice:
Factoring Polynomials into Binomials:
1)
x2 + 2x + 1
2)
x2 - 9
3)
x2 + 5x + 6
4)
x2 + 3x - 40
5)
x2 – 6x + 8
6)
x2 – 11x + 18
7)
x2 -7x + 12
8)
x2 + 12x + 32
DEV 108 Notes
Revised Winter 2007
p. 12
9)
x2 – x – 30
10)
x2 + 10x + 9
11)
x2 + x – 42
12)
x2 – 10x + 12
13)
x2 – 81
14)
x2 + 3x - 10
DEV 108 Notes
Revised Winter 2007
p. 13