Download 6-1 and 6-2 Solving Inequalities

6-1 and 6-2 Solving Inequalities
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
For this lesson you will need a
ruler
and a colored pencil.
The graph of an inequality in one variable is the set of
points on a number line that represent all solutions of the
inequality.
x  4 endpoint
•
O
ray
If the endpoint on the graph is not a solution, draw an open
dot.
If the endpoint on the graph is a solution, draw a solid dot.
Then draw an arrowhead to show that the graph continues to
infinity.
What is the name
for the geometric
figure that
represents the
solution?
Graphing an Inequality in One Variable
1. Write inequality.
Rewrite with variable first.
7 > x
2. Draw a line (use arrowheads).
•
3. Draw open or solid dot and
label the endpoint.
4. Draw the ray in the direction
of the inequality symbol.
You do NOT
need to
draw in the
tick marks.
x < 77
A solution of an inequality is a value for the variable that
makes the inequality true.
You can add or subtract inequalities just like you add or
subtract equations.
To solve an inequality isolate the variable on one side of
the inequality symbol. Follow the basic rule:
Whatever you do to one
side of the inequality sign,
you must also do to the
other side of the inequality
sign.
Solve x – 6 > –14. Then graph the solution.
Write the inequality.
Isolate the variable using
inverse operations.
Graph.
x  6   14
6
6
x  –88
•
Solve x – 6 > –14. Then graph the solution.
Write the inequality.
Isolate the variable using
inverse operations.
Graph.
x  6   14
6
6
x  8
•
–8
How many
solutions are there
to an inequality
problem?
Infinite
Solve x – 6 > –14. Then graph the solution.
Write the inequality.
Isolate the variable using
inverse operations.
Graph.
x  6   14
6
6
x  8
•
–8
The authors of your text use
set-builder notation to write
the solution. {x x   8}
The set of all numbers x such
that x is greater than or equal
to -8.
Solve. Then graph the solution.
Example 1
13  c  45
Example 2
3  m  5
Example 3
81.300  r  107.325
1 3
m 
4 8
Example 4
Example 5
5x  7  6x
Example 6 Write an inequality and then solve.
Forty is no greater than the difference of a
number and two.
Solve. Then graph the solution.
Example 1
13  c  45
 45
 45
58  c
c  558
8
•
Example 2
3  m  5
5
5
8  m
m  88
O
Did you
draw an
open dot?
Solve. Then graph the solution.
Example 3 81.300  r  107.325
 81.300
 81.300
26.025
r  26
.025
•
1 3

4 8
1
2
1


4 8
4
1
m
8
Example 4 m 
•
Solve. Then graph the solution.
Example 5 5x  7  6x
 5x
 5x
7x
x  77
O
Ex. 6 Forty is no greater than the difference of a number
<
and two.
40  n  2
2
2
42  n
n  42
Multiplication and Division Properties of Inequalities
If you multiply or divide each side of an inequality by a
positive number, the direction of the inequality symbol
is unchanged.
3x  12
3x  12
3
3
x  4
If you multiply or divide each side of an inequality by a
negative number, the direction of the inequality symbol
is reversed.
 3x  12
 3x / 12
3  3
x4
m
Solve
 6 .
3
Write the inequality.
Use inverse property.
When you multiply or
divide by a negative,
reverse inequality symbol.
m
 6
3
 3 m /
 6 3
3
>
m  18
Ex. 7
m
 2
2
Ex. 9
1
1
3 m2
4
2
Solve.
Ex. 8
42   6w
Ex. 10  2.8d  9.8
Write an inequality and then solve.
Ex. 11 Negative three eighths times a number is
greater than or equal to 12. Find the number.
Ex. 12 Two and one half times a number is less than
one and one fifth. Find the number.
Ex. 7 m  2
2
2 m  2 2
2
m  4
1
1
3 m2
4
2 2
5 4 
 4  13
  m  
2  13 
 13  4
10
m
13
Ex. 9
Solve.
Ex. 8
42   6w
42 
/  6w
6 > 6
7  w
w  7
Ex. 10  2.8d  9.8
 2.8d / 9.8
 2.8  2.8
>
d  3.5
Ex. 11 Negative three eighths times a number is greater
>
than or equal to 12.
3
 n  12
8
4
 8
 8 3
 12


 n /
3

 3 8 < 
n  32
Ex. 12 Two and one half times a number is less than one
<
and one fifth.
1
1
2 n 1
2
5
 2 5 n  6  2 
 
 
55
 5 2
12
n
25
6-A2 Pages 297 # 20–33, and
Pages 305 #17-28.
Inductive Reasoning is making conclusions on patterns
you observe.
Look for the pattern that results
from dividing each side of an
inequality by an integer.
6 3
6 2
2 < 4
3< 6
6 < 12
6 1
The inequality symbol reverses
6   1  6 >  12
direction when you divide by a
6   2  3 >  6
negative number.
6   3  2 >  4
•
-12 -10
-8
•
-6
An inequality
symbol points to
the smaller
number.
12  3
12  2
12  1
12   1
12   2
12   3