Download Solving Inequalities PPT (notes)

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Solving Inequalities
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An inequality is like an equation,
but instead of an equal sign (=) it
has one of these signs:
< : less than
≤ : less than or equal to
> : greater than
≥ : greater than or equal to
What do Inequalities mean?
• A mathematical sentence that uses
one of the inequality symbols to
state the relationship between two
quantities.
Graphing Inequalities
• When we graph an inequality on a number
line we use open and closed circles to
represent the number.
<
<
Plot an open circle
≤
≥
Plot a closed circle
x<5
means that whatever value x
has, it must be less than 5.
Try to name ten numbers that
are less than 5!
Numbers less than 5 are to the left
of 5 on the number line.
-25 -20 -15 -10 -5
0
5
10 15 20 25
• If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right.
• There are also numbers in between the integers, like
2.5, 1/2, -7.9, etc.
• The number 5 would not be a correct answer,
though, because 5 is not less than 5.
x ≥ -2
means that whatever value x
has, it must be greater than or
equal to -2.
Try to name ten numbers that
are greater than or equal to
-2
Numbers greater than -2 are to the
right of -2 on the number line.
-25 -20 -15 -10 -5
0
5
10 15 20 25
-2
• If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right.
• There are also numbers in between the integers,
like -1/2, 0.2, 3.1, 5.5, etc.
• The number -2 would also be a correct answer,
because of the phrase, “or equal to”.
Homework
Solving an Inequality
• Solve much like you would an
equation.
• Always undo addition or subtraction
first, then multiplication or division.
• Remember whatever is done to one
side of the inequality must be done to
the other side. The goal is to get the
variable by itself.
Properties to Know for
Solving Inequalities
Addition and Subtraction
• Adding c to both sides of an inequality just shifts everything along,
and the inequality stays the same.
• If a < b, then a + c < b + c
Example: Alex has less coins than Billy.
• If both Alex and Billy get 3 more coins each, Alex will still have less
coins than Billy.
• Likewise:
• If a < b, then a − c < b − c
• If a > b, then a + c > b + c, and
• If a > b, then a − c > b − c
• So adding (or subtracting) the same value to both a and b will not
change the inequality
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Properties to Know for
Solving Inequalities
Multiplication and Division
When we multiply both a and b by a positive number, the inequality stays the same.
But when we multiply both a and b by a negative number, the inequality swaps over!
Notice that a<b becomes b<a after multiplying by (-2)
But the inequality stays the same when multiplying by +3
Here are the rules:
If a < b, and c is positive, then ac < bc
If a < b, and c is negative, then ac > bc (inequality swaps over!)
A "positive" example: Alex's score of 3 is lower than Billy's score of 7.
a<b
If both Alex and Billy manage to double their scores (×2), Alex's score will still be lower than Billy's
score.
2a < 2b
But when multiplying by a negative the opposite happens:
But if the scores become minuses, then Alex loses 3 points and Billy loses 7 points
So Alex has now done better than Billy!
-a > -b
The same is true for division – flip the sign of the inequality if dividing by a negative number
Solve an Inequality
w+5<8
-5
-5
w< 3
-25 -20 -15 -10 -5
0
5
All numbers less
than 3 are
solutions to this
problem!
10 15 20 25
1 step Examples
8 + r ≥ -2
-8
All numbers greater than-10
(including -10)
-8
r ≥ -10
-25 -20 -15 -10 -5
0
5
10 15 20 25
1 step Examples
2x > -2
2
2
x > -1
-25 -20 -15 -10 -5
0
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All numbers
greater than -1
make this problem
true!
10 15 20 25
2 step Examples
2h + 8 ≤ 24
-8
-8
2h ≤ 16
2
2
All numbers less
than 8 (including 8)
h≤8
-25 -20 -15 -10 -5
0
5
10 15 20 25
Be Aware of Cases Involving Multiplying and
Dividing Inequalities with Negative Numbers
• Multiplication Example
• Division Example
One More Case
• Solve Inequalities with the variable on both
sides
Your Turn….
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Solve the inequality and graph the answer.
1. x + 3 > -4
x > -7
2. 6d > 24
d>4
3. 2x - 8 < 14
x < 11
4. -2c – 4 < 2
*c ≥-3 noticed in this problem you had to flip the inequality
Summarize:
Solving Inequalities
Be sure to know the properties
affecting inequalities.
• Addition and Subtraction: Adding(or
subtracting) c to both sides of an inequality
just shifts everything along, and the
inequality stays the same.
• If a < b, then a + c < b + c
• If a < b, then a - c < b - c
Be sure to know the properties
affecting inequalities.
• Multiplication and Division: When we multiply
both a and b by a positive number, the inequality stays
the same.
• But when we multiply both a and b by a negative number,
the inequality swaps over!
Notice that a<b becomes b<a after multiplying by (-2)
But the inequality stays the same when multiplying by +3
• The same is true for division – flip the sign of the
inequality if dividing by a negative number
Let’s look at some
REAL-LIFE APPLICATION
PROBLEMS
Real-Life Application
You are taking a history
course in which your grade
is based on six 100 point
tests. To earn an A in class,
you must have a total of at
least 90%. You have
scored an 83, 89, 95, 98,
and 92 on the first five
tests. What is the least
amount of points you can
earn on the sixth test in
order to earn an A in the
course?
Hint: 90 x 6
90% = 540 pts.
83+89+95+98+92= 457
457 - 457 + T ≤ 540 - 457
T ≤ 83
Example 2
To play a board
game, there must
be at least 4 people
on each team. You
divide your friends
into 3 groups.
Write and solve an
inequality to
represent the
number of friends
playing the game.
• f/3 ≥ 4
• f/3 ▪ 3 ≥ 4 ▪ 3
• F ≥ 12
Example 3:
You budget $50 a
month for your cell
phone plan. You pay
$45 for your minutes
and 250 text messages.
You are charged an
extra $0.50 for picture
messages. Write and
solve an inequality to
find the number of
picture messages you
can send without
going over your
budget.
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0.50 x +45 ≤ 50
0.50 x +45 -45 ≤ 50 -45
0.50 x ≤ 5
0.50 x / 0.50 ≤ 5 /0.50
x ≤ 10
Go Forth and Prosper!
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