Transcript 7-1: Solving Inequalities by Using Addition and Subtraction

7-1: Solving Inequalities by Using Addition and Subtraction
OBJECTIVE:
You will be able to solve inequalities by using addition and subtraction.
This section is essentially the same as solving equations.
The major difference is that instead of finding one single value which
makes an equation true, the answers to inequalities is a set of answers.
You need to recall the work done in Chapter 2 involving inequalities.
Specifically, the significance of which way they point and how their number-line
graphs appear.
If you understood how to solve equations in Chapter 3, these inequalities
will be easy.
If you did not understand Chapter 3, you need to go back and review it.
© William James Calhoun, 2001
7-1: Solving Inequalities by Using Addition and Subtraction
The symbols and their meanings again:
Inequalities
<
•less than
•fewer than

>
•greater than
•more than
•at most
•no more than
•less than or
equal to

•at least
•no less than
•greater than
or equal to
In solving inequalities by using addition and subtraction, the rule is the
same as solving equations:
YOU CAN ADD OR SUBTRACT ANYTHING AS LONG AS YOU
ADD OR SUBTRACT THE SAME THING ON BOTH SIDES.
With that, we will start our first example.
© William James Calhoun, 2001
7-1: Solving Inequalities by Using Addition and Subtraction
EXAMPLE 1: Solve 29 + g  68.
29 + g  68
-29
-29
g  39
The questions are the same.
What is the letter?
g
What is on the same side as the g?
positive 29
How are the 29 and the g combined?
addition
Get rid of the 29 by…
subtracting 29 from both sides.
Do it.
This is the answer.
The book gives its answers in set-builder notation.
In set-builder notation, the answer would look like this:
{g | g  39}
This is read, “g such that g is less than or equal to 39.”
If we were to graph this solution set, it would appear like this:
34
35
36
37
38
39
40
41
42
43
44
45
46
47
Notice the full circle at the point and the arrow showing the solution set goes on
forever.
© William James Calhoun, 2001
7-1: Solving Inequalities by Using Addition and Subtraction
Example 2: Solve 13 + 2z < 3z - 39. Then graph the solution.
13 + 2z < 3z - 39
- 2z -2z
13 < z - 39
+39
+39
52 < z
{z | z > 52}
What is the letter?
z
What is on the same side as z?
which z?
We must get all the z’s on the same side.
For now, make sure the resulting z’s are positive.
Because of this, we need to move the 2z by…
subtracting 2z from both sides.
Do it.
Now, what is on the same side as the z?
negative 39
Get rid of it by…
adding 39 to both sides.
Now graph it.
49
50
51
52
53
54
We will talk about negative variables and their implications on inequalities next
section.
© William James Calhoun, 2001
7-1: Solving Inequalities by Using Addition and Subtraction
EXAMPLE 3: Alvaro, Chip, and Solomon have earned $500 to buy
equipment for their band. They have already spent $275 on a used
guitar and a drum set. They are now considering buying a $125
amplifier. What is the most they can spend on promotional materials
and T-shirts for the band if they buy the amplifier?
You cannot spend more than what you have, so the amount the group spends MUST be less than or equal
to the amount they have.
What did they start with?
what they spend  what they have $500
Let “p” be the amount of promo materials.
275 + 125 + p  500
What will be the amount they spend?
$275 + $125 + p
400 + p  500
Combine like terms on the left-hand side.
-400
-400
What is the letter?
p
p  100
What is on the same side as p?
400
If the band purchases the amplifier, they
How to get rid of it?
will have at most $100 to spend on
subtract 400 from both sides
promo materials.
© William James Calhoun, 2001
7-1: Solving Inequalities by Using Addition and Subtraction
EXAMPLE 4: Write an inequality for the sentence below. Then solve
the inequality and check the solution.
Three times a number is more than the difference of twice that number and three.
3
*
Rewrite it.
Solve it.
x
>
2*
x
-
3
3x > 2x - 3
-2x -2x
x > -3
To check it, plug in any number larger than -3 and see if it makes a true statement.
0 > -3
10 > -3
-2 > -3
100 > -3
3(0) > 2(0) - 3 3(10) > 2(10) - 3 3(-2) > 2(-2) - 3 3(100) > 2(100) - 3
0 > -3
30 > 17
-6 > -7
300 > 197
True
True
True
True
© William James Calhoun, 2001
7-1: Solving Inequalities by Using Addition and Subtraction
HOMEWORK
Page 388
#17 - 35 odd
© William James Calhoun, 2001