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Algebra Chapter 5 Notes
Solving and Graphing Linear Inequalities
5.1 Solve Inequalities Using Addition and Subtraction
Graph of an inequality: ________________________________________________________________
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Equivalent Inequalities: ________________________________________________________________
Example 1: Write and graph an inequality that describes the situation.
a. An infant car seat is designed for babies and toddlers weighing less than 40 pounds.
b. A sign on a store display says items are $4 or higher.
c. You must be at least 5 years old to go to kindergarten in Pennsylvania.
Example 2: Write an inequality represented by the graph.
Example 3: Solve the following inequalities. Graph your solution.
x – 1.3 < 2.8
x – 7  –3
5.1 > y – 2.7
Example 4: Solve the following inequalities. Graph your solution.
13  x + 4
z + 9 < –1
6  w + 1.5
5.2 Solve Inequalities Using Multiplication and Division
Multiplication Property of Inequality: _____________________________________________________
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Division Property of Inequality: _________________________________________________________
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Example 1: Solve the inequalities. Graph your solutions.
x
> -3
7
x
≤5
2
Example 2: Solve the inequalities.
6x > –36
–3x ≤ 9
18 ≥ 9x
6x < 12
Example 3: A library has $180 to buy new books. The books cost $9 each. Write and solve an inequality
to find the possible number of books that can be bought for the library.
5.3 Solve Multi-Step Inequalities
Example 1: Solve and graph.
–4x + 3 > 15
7x + 8 > 22
–7 ≥ –2x + 9
Example 2: Solve the inequalities.
–
1
(x + 12) < 5
3
9x + 2 < 5x – 18
Example 3: Solve the inequality, if possible.
5(3x – 2) < 15x + 7
9 – 28x > 4(5 – 7x)
Example 4: You can work at most 24 hours per week as a nurse’s aide. So far this week you have
worked 7 hours. If the remaining shifts for the week are each 4 hours long, how many possible full shifts
can you work?
5.4 Solve Compound Inequalities
Compound inequality: _________________________________________________________________
___________________________________________________________________________________
Example 1: Translate the verbal phrases into an inequality. Then graph the inequality.
a. All real numbers that are less than or equal to 7 or greater than or equal to 10.
b. All real numbers that are greater than –1 and less than or equal to 1.
c. All real numbers that are less than –3 or greater than 0.
d. All real numbers that are less than 9 and greater than or equal to 7.
Example 2: Solve the inequalities. Graph your solutions.
7 ≤ x – 4 ≤ 12
30 ≥ –7x – 12 > 16
28 ≤ 4(2x – 3) ≤ 68
Example 3: Solve the inequalities. Graph your solutions.
3x + 4 < 16 or 5x – 12 > 13
3x + 8 > 7x – 12 or 9(x – 2) > 8x – 9
Example 4: The eggs of a Rocky Mountain Tailed frog can survive in streams where the temperatures
range from 5  C to 18  C . Write a compound inequality that describes the possible stream temperatures
(in degrees Fahrenheit) for egg survival. Solve the inequality. Then graph your solution. Identify three
possible stream temperatures (in degrees Fahrenheit) for egg survival.
5.5 Solve Absolute Value Equations
Absolute value equation: _______________________________________________________________
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Absolute deviation: ___________________________________________________________________
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Examples 1 and 2: Solve the equations.
|x| = 3
|2x – 1| = 7
Example 3: Solve the equations.
1
|3x – 6| + 7 = 13
2
2|x – 1| –5 = 9
Example 4: Solve the equations, if possible.
|2x – 1| + 4 = 3
5|x – 4| + 11 = 8
Example 5: A volleyball league is preparing a two minute radio ad to announce tryouts. The ad has an
absolute deviation of 0.05 minutes. Find the minimum and maximum acceptable times the radio ad can
run.
5.6 Solve Absolute Value Inequalities
Solving Absolute Value Inequalities: _____________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
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Example 1: Solve the inequality. Graph your solution.
|x| ≤ 5
|x| ≥ 1.5
Example 2: Solve the inequalities. Graph your solutions.
|–x + 2| < 7
|8x + 5| > 17
Example 3: Solve the inequalities. Graph your solutions.
|5x – 1| – 4 ≥ 7
4|x + 7| – 3 ≤ 5
5.7 Graph Linear Inequalities in Two Variables
Linear inequality in two variables: _______________________________________________________
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Graph of an inequality in two variables: ___________________________________________________
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Example 1: Tell whether the ordered pair is a solution of the inequality.
1
x – 3y  8; (10, –3)
2
3x – y > 7; (4, 3)
Example 2: Graph the inequalities 3x – y  1 and x + y  –2.
Example 3: Graph the inequalities y >
1
and x  1.
2