Transcript less than or equal to

Winter Warm up
• There are 25 students in my class. 17 said they would go
snow skiing this year, 20 said they would go to Universal
Studios and 4 would not do either. How many students will
do both this semester?
Snow skiing
Universal Studios
Solution
Snow skiing
1
16
Universal Studios
4
4
Neither
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<
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Solving Inequalities
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Lesson 4-5
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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
Review of Inequality Signs
> greater than
< less than
 greater than or equal
 less than or equal
How to graph the solutions
> Graph any number greater than. . .
open circle, line to the right
< Graph any number less than. . .
open circle, line to the left
 Graph any number greater than or equal to. . .
closed circle, line to the right
 Graph any number less than or equal to. . .
closed circle, line to the left
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”.
Solving an Inequality
• Follow the same rules and steps that
we used to solve an equation.
• Always undo addition or subtraction
first, then multiplication.
• 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.
Solve the inequality:
x+4<7
-4 -4
x < 3
●Subtract 4 from each side.
●Keep the same inequality sign.
●Graph the solution.
• Open circle, line to the left.
-25 -20 -15 -10 -5
0
5
10 15 20 25
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
More 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
More Examples
2x > -2
2
2
x > -1
-25 -20 -15 -10 -5
0
5
All numbers
greater than -1
make this problem
true!
10 15 20 25
More 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
There is one special case.
● Sometimes you may have to reverse the
direction of the inequality sign!!
● That only happens when you
multiply or divide both sides of the
inequality by a negative number.
Example:
-3y + 5 >23
-5 -5
-3y > 18
-3 -3
y < -6
●Subtract 5 from each side.
●Divide each side by negative 3.
●Reverse the inequality sign.
●Graph the solution.
•Open circle, line to the left.
-6
0
Your Turn….
•
Solve the inequality and graph the answer.
1.
2.
3.
4.
x + 3 > -4
6d > 24
2x - 8 < 14
-2c – 4 < 2
1.
2.
3.
4.
x > -7
d>4
x < 11
c < -3
Try these:
5. 2x+3>x+5
6. - c - 11>23
7. 3(r-2)<2r+4
5. x > 2
6. c < 34
7. r < 10
Homework
• Complete the odds on the 5.1 and 5.2
workseets