Download Chapter 2: Linear Equations and Inequalities - 1 -

Chapter 2: Linear Equations and Inequalities
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2.6: One-Variable Linear Inequalities
Homework and Future Dates:
Homework 1: Linear Inequalities Worksheet
Homework 2: Barron’s Book p. 66 (1-6, 13-16) Show all work
Homework 3: Barron’s Book p. 66 (7 - 12) Show all work
Exam 1 Retest: Monday, 26 October either 7:30 AM or 2:50 PM in E112. This grade
with be averaged with your 1st Exam 1 grade.
Next Test (Exam 3):
A Period: 1 November
E Period: 2 November
Inequality Symbols:
 less than
 greater than
 less than or equal to  greater than or equal to
 not equal to
The solution of an inequality includes any values that make the inequality true.
Solutions to inequalities can be graphed on a number line using open and closed dots.
Open dots vs. Closed dots
When the inequality sign does not contain
When the inequality sign contains includes
an equality bar beneath it, the dot is open.
an equality bar beneath it, the dot is closed,
or shaded in.
Graph of x  1
-3
-2
-1
0
1
Graph of x  1
2
3
Lesser
-3
Greater
-2
-1
0
1
2
0
1
2
3
Greater
Graph of x  1
-3
3
Lesser
-1
Lesser
Graph of x  1
-3
-2
Greater
-2
-1
0
Lesser
1
2
3
Greater
Graph of x  1
-3
-2
-1
0
1
Lesser
2
3
Greater
To Graph an Inequality, you must do the following:
________________________________________________________________________
Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be
sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with
a parent/ guardian signature.
Chapter 2: Linear Equations and Inequalities
2.6: One-Variable Linear Inequalities
1. Create a number line
2. Place at least three numbers on the line (the solution number, zero and a third
number)
3. If x is on the left of the inequality, move on to step 4. If x is on the right, flip the
inequality.
4. Place an open or closed circle on the number
5. Shade to the side of the number that contains the solution set (draw your arrow in
the direction of the inequality sign)
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Example 1
Graph
x  5
The statement x  5 says that x can take any value greater than -5, but x cannot equal
-5 itself. We show this on a graph by placing an open circle at -5 and drawing an arrow
to the right. The open circle at -5 shows that -5 is not part of the graph.
Solution
Example 2
Graph
3 x
The statement
3  x means the same as ___________________.
________________________________________________________________________
Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be
sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with
a parent/ guardian signature.
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Chapter 2: Linear Equations and Inequalities
2.6: One-Variable Linear Inequalities
Solution
Work on these examples with your groups:
On the number line provided, graph each inequality:
(a)
x3
(b)
x4
(c)
4 x
________________________________________________________________________
Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be
sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with
a parent/ guardian signature.
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Chapter 2: Linear Equations and Inequalities
2.6: One-Variable Linear Inequalities
(d)
0x
Solutions:
(a)
x3
(b)
x4
(c)
 4  x Flip: x  4
(d)
0  x Flip: x  0
________________________________________________________________________
Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be
sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with
a parent/ guardian signature.
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Chapter 2: Linear Equations and Inequalities
2.6: One-Variable Linear Inequalities
Writing Equivalent Inequalities
Just like when we solve equation, most of the time, we do the same operation to both
sides of an inequality…
An equivalent inequality results when
 The same quantity is added to, or subtracted from, both sides
 Both sides are multiplied or divided by the same positive quantity.
But inequalities have a tick when doing multiplication and division with a negative
number….
 Both sides are multiple or divide by the same negative quantity and the direction
of the inequality sign is reversed.
For example
12  8
12
8

4 4
 3  2
________________________________________________________________________
Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be
sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with
a parent/ guardian signature.
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Chapter 2: Linear Equations and Inequalities
2.6: One-Variable Linear Inequalities
Solving a Linear Inequality
Solve an inequality the same way that you solve an equality, except when you multiply
or divide both sides by a negative number, you must turn the inequality sign around.
8  3x  17
8
8
 3x
9

3 3
x  3
Example 1
Find the solution set of: 25  4x  13
Subtract 25:
Divide each side by -4 and flip the inequality sign:
Example 2
Find and graph the solution set of
1  2x  x  6
________________________________________________________________________
Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be
sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with
a parent/ guardian signature.
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Chapter 2: Linear Equations and Inequalities
2.6: One-Variable Linear Inequalities
Directions: Solve and graph the following inequalities on a separate sheet of paper.
Homework: Barron’s Book p. 66 (1-6, 13-16) Show all work
________________________________________________________________________
Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be
sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with
a parent/ guardian signature.
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Chapter 2: Linear Equations and Inequalities
2.6: One-Variable Linear Inequalities
Union of Sets
Example 1
Graph the solution set for x 1  2x  9 x 3x  1  7.
Step 1: Solve each equations:
1  2x  9
3x  1  7
Step 2: Graph the solution set of x 1  2x  9, which is x  4
Step 3: Graph the solution set of x 3x  1  7, which is x  2
Step 4: Graph the intersection of the two solution sets.
Step 5: Describe this interval with an inequality.
________________________________________________________________________
Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be
sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with
a parent/ guardian signature.
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Chapter 2: Linear Equations and Inequalities
2.6: One-Variable Linear Inequalities
Directions: Solve and graph the following inequalities on a separate sheet of paper.
Homework: Barron’s Book p. 66 (7 - 12) Show all work
________________________________________________________________________
Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be
sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with
a parent/ guardian signature.
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