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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.
O•
What is the name for the
geometric figure that
represents the solution?
The graph of an inequality in one variable is the set of points on a number line that represent all solutions of the inequality.
ray 4x 4
If the endpoint on the graph is a solution, draw a solid dot.
If the endpoint on the graph is not a solution, draw an open dot.
Then draw an arrowhead to show that the graph continues to infinity.
endpoint
Graphing an Inequality in One Variable
1. Write inequality.
7
•x < 7
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.
7 > x Rewrite with variable first.
You do NOT need to draw in
the tick marks.
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. 14 6 x
Isolate the variable using inverse operations.
6 6 8 x
Graph.
–8
•
Solve x – 6 > –14. Then graph the solution.
Write the inequality. 14 6 x
Isolate the variable using inverse operations.
6 6 8 x
Graph.–8•
How many solutions are there to an inequality problem?
Infinite
Solve x – 6 > –14. Then graph the solution.
Write the inequality. 14 6 x
Isolate the variable using inverse operations.
6 6 8 x
Graph.–8•
The authors of your text use set-builder notation to write the solution. }8 x x{
The set of all numbers x such that x is greater than or equal to -8.
Solve. Then graph the solution.
Example 2
5 3 m
Example 1
45 c 31
Example 3
325.107r300.81
Example 4 8
341
m
Example 5
x67x5
Forty is no greater than the difference of a number and two.
Example 6 Write an inequality and then solve.
Solve. Then graph the solution.
8 m
Example 2
5 3 m5 5
m 8 8
O
Did you draw an
open dot?
Example 1
45 c 31 45 45
c 85
•85 c 58
Example 3 300.81 300.81
025.26r
•26.025
Solve. Then graph the solution.
325.107r300.81
Example 4
41
41
81
m
•
83
41
m
82
81
Example 5
x67x5 x5 x5
7 x 7
Solve. Then graph the solution.
x 7
O
Ex. 6 Forty is no greater than the difference of a number and two. 2n40
2
42n
2<
n42
If you multiply or divide each side of an inequality by a negative number, the direction of the inequality symbol is reversed.
123x 3 3
4x
12x3 3 3
4x
If you multiply or divide each side of an inequality by a positive number, the direction of the inequality symbol is unchanged.
Multiplication and Division Properties of Inequalities
/
123x
123x
>
Solve
Write the inequality.
63
m
Use inverse property. 3 3
18m
.63
m
When you multiply or divide by a negative, reverse inequality symbol.
/
63
m
Solve. 2
2m Ex. 7 Ex. 8 w6 42
Ex. 921
2m41
3 Ex. 10
8.9d8.2
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.
Write an inequality and then solve.
Solve.
22m 2 2
4m
22m Ex. 7 Ex. 8
>
w6 42 6 6
w7 7w
/w6 42
Ex. 9
25
m413
134
134
1310
m
21
2m41
3 2
>
Ex. 10 8.9d8.2
8.2 8.2
5.3d
/
8.9d8.2
Ex. 11 Negative three eighths times a number is greater than or equal to 12.
12n83
38
32n
38
>
Ex. 12 Two and one half times a number is less than one and one fifth.
51
1n21
2
52
2512
n
52
<
12n83
</
4
56
n25
6-A2 Pages 297 # 20–33, and
Pages 305 #17-28.
36Look for the pattern that results from dividing each side of an inequality by an integer.
The inequality symbol reverses direction when you divide by a negative number.
312<
26 212<
16 112< 16 112 >
26 212 > 36 312 >
-12•
-10 -8 -6•
2 43 6
6 126 12
3 6
2 4
Inductive ReasoningInductive Reasoning is making conclusions on patterns you observe.
An inequality symbol points to the smaller
number.
