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M1U1D4 Warm Up:
Solve for h:
A = ½ bh
h = 2A/b
Homework Check:
Document Camera
QUIZ:
Return and Discuss IF all make-ups complete.
M1U1D4
Solving and Graphing
Linear Inequalities
Objective:
To create equations with one
variable and use them to solve
multi-step problems.
What is an inequality?
•
• It is a range of values,
rather than ONE set number
An algebraic relation showing that a
quantity is greater than or less than
another quantity.
Speed limit:
55  x  75
Symbols




Less than
Greater than
Less than OR EQUAL TO
Greater than OR EQUAL TO
Solutions….
You can have a range of answers……
-5 -4 -3 -2 -1 0
1
2
All real numbers less than 2
x< 2
3
4
5
Solutions continued…
-5 -4 -3 -2 -1 0
1
2
All real numbers greater than -2
x > -2
3
4
5
Solutions continued….
-5 -4 -3 -2 -1 0
1
2
3
4
5
All real numbers less than or equal to 1
x 1
Solutions continued…
-5 -4 -3 -2 -1 0
1
2
3
4
5
All real numbers greater than or equal to -3
x  3
Did you notice,
Some of the dots were solid
and some were open?
x2
-5 -4 -3 -2 -1
0
1
2
3
4
5
-5 -4 -3 -2 -1
0
1
2
3
4
5
x 1
Why do you think that is?
If the symbol is > or < then dot is open because it can not be
equal.
If the symbol is  or  then the dot is solid, because it can be
that point too.
Ex 1: Write and Graph
a Linear Inequality
Sue ran a 10-K race in 5 minutes. Write an inequality
to describe the average speeds of runners who were
faster than Sue. Graph the inequality.
Faster average speed >
s2
Distance
10
s
5
Sue’s Time
-5 -4 -3 -2 -1 0
1
2
3
4
5
Solving an Inequality
Solving a linear inequality in one variable is much like
solving a linear equation in one variable. Isolate the
variable on one side using inverse operations.
Solve using addition:
x–3<5
Add the same number to EACH side.
x 3  5
+3
+3
x<8
Solving Using Subtraction
Subtract the same number from EACH side.
x  6  10
-6
-6
x4
Ex 2: Using Subtraction…
x 5  3
Solve and graph
the solution.
-5 -4 -3 -2 -1 0
x > -2
1
2
3
4
5
Ex 3: Using Addition…
2  n4
Solve and graph the
solution.
-5 -4 -3 -2 -1 0
n<2
1
2
3
4
5
THE TRAP…..
When you multiply or divide each side of
an inequality by a negative number, you
must reverse the inequality symbol to
maintain a true statement.
Solving using Multiplication
Multiply each side by the same positive number.
1
(2)
x  3 (2)
2
x6
Solving Using Division
Divide each side by the same positive number.
3x  9
3
3
x3
Solving by multiplication of a
negative #
Multiply each side by the same negative number
and REVERSE the inequality symbol.
(-1)
 x  4 (-1)
Multiply by (-1).
See the switch
x  4
Solving by dividing by a
negative #
Divide each side by the same negative
number and reverse the inequality symbol.
 2x  6
-2
-2
x  3
Ex 4: Solve the inequality.
a)
2x-3<8
+3 +3
2x<11
2 2
11
x<
b)
 3x  7  13
 3x  6
x  2
2
Flip the sign after dividing
by the -3!
Graphing Linear Inequalities
• Remember:
< and > signs will have an open dot o
 and  signs will have a closed dot 
graph of
4 5
6
7
11
x
2
graph of x  2
-3 -2 -1 0
Example 5: Solve and graph the solution.
7 x  9  10x 12
9  3x 12
21  3x
7 x
x<7
6
7
8
9
Classwork:
M1U1D4 CW Handout
#1: n + 13 > 23
Homework:
M1U1D4 HW Handout
#1: -22 > -11 + x