Chapter 6 Solving Linear Inequalities
Name:________________
Introduction to Inequalities:
Example 1: Graph the following inequalities on a number line.
a)
4x

b)
x  1
d)
5x

c)
x  3


Rules for graphing inequalities:
First, if the variable is on the right, ____________________________________.
Ex:
Then:

Use an ____________ circle on the number line for _____________.

Use a _____________ circle on the number line for _____________.
Finally, shade in the direction of the sign:


For > or  , shade to the ______________.
For < or  , shade to the ______________.


Examples 2: Write an inequality from the following graphs.
a)
b)
c)
Section 6 – 1: Solving Inequalities by Addition and Subtraction
Solving Inequalities are similar to solving equations except:
o You will get an infinite number of answers (in most cases) which you can
represent on a number line or as an inequality statement
Words Used to Describe Inequalities
<

>



Example 1: Translate the sentence into inequalities and represent the solution with a graph.
a) A number is less than or equal to 3
b) A number is less than -1
c) A number is greater than or equal to 2
d) A number is greater than 4
e)
x is at least 4
f)
x is at most -3
g)
a number does not exceed 2
h)
a number is no greater than -4
i)
x is no less than 4
j)
5 is less than a number.
Solving an inequality by addition or subtraction is the same as solving equations.
Example 2: Solve the following inequalities and graph the solution.
1)

x  8 18
2)

x 1  6
3)

7  x  8
4)

x  6  10
6 – 2: Solving Inequalities by Multiplication and Division
Solving an inequality using multiplication and division is similar to solving an equation.
Example 1: Solve the following inequalities and graph the solution.
a)
6g  144
b)

2
v 6
3

However, there is one difference between solving equations and inequalities.
2x  10
Divide both sides by -2.
2x 10

2
2

x  5

Substitute a number that is greater than -5 into the original inequality. For example, x = 0.

20 10
0  10
But this is not true, so something is wrong in the solution.


When you multiply or divide both sides of an inequality by a negative number, ______
________________________________________________________________________.
c)
14d 84


f)
3
 q  33
4

e)

b
5
10
d)

r
 7
7

6 – 3: Solving Multi-Step Inequalities
Steps to solving multi-step Inequalities:
1)
Distribute.
2)
Combine like terms.
3)
Simplify using addition and subtraction.
4)
Simplify using multiplication and division.
5)
If the variable is on the right, switch it to the left and reverse the direction of the
inequality.
6)
Test your solution by substituting a value into the original inequality.
Remember:
REVERSE the inequality sign when dividing or multiplying by a ______________.
Example 1: Solve the following inequalities and graph the answers.
a)
3t  6  3

2 
d)
5(2h  6)  4h

c)

d
 23
5
b)
2v  3
7
5
Example 2: Special cases.
e)
3  3(b  2)  13  3(b  6)

Summarizing:
-
If a simplified inequality results in a ________ statement, there is _________ solution.
-
If a simplified inequality results in a ________ statement, the solution is
_______________ numbers.
Example 3: Write an inequality for the following statements, solve and graph the solution.
a) The sum of a number and 13 is at least 27.
b) Thirty is no greater than the sum of a number and -8.
c) Twenty-four is at most a third of a number.
d) Two thirds of a number plus eight is greater than twelve.
Example 4: Application of inequalities.
1)
The charge per mile for a compact car rental at Great Deals Rentals is $0.25.
There is also a rental fee of $50. Mr. Jones has a budget of $150 for a car rental.
How many miles can Mr. Jones drive and not exceed his budget?
a. Write an inequality to represent the situation. Define the variable. Then solve
the inequality.
b. State the answer.
2)
Find all sets of two consecutive positive even integers who sum is no greater than
18.
a. Write an inequality to represent the situation. Define the variable. Then solve
the inequality.
b. State the answers.
6 – 4: Compound Inequalities
Compound Inequality:
Conjunctions – inequalities joined by the word ________________.
To be a solution, a number has to satisfy ______________ inequalities.
Example 1:
a)


c)

x  5 and x  8
b)

3  x  5

d)

x  4 and x  4
4x7
e)
5  2x  4  2

f)
3y 12  6  y  3y 18

Disjunctions – inequalities joined by the word ______________
To be a solution, a number can satisfy ____________ inequality.
Example 2:
a)


c)

x  5 or x  1
x  5 or x  9

b)

x  6 or x  3

3x  4  19 or 7x  3  18
d)


e)

1 x  3 or x  4

Example 3: Application of Compound Inequalities
a)
Each type of fish thrives in a specific range of temperatures. The optimum
temperature sharks range from 18o C to 22o C, inclusive. Write an inequality to
represent temperature where sharks will NOT thrive.
6.5 - Absolute Value (Solving Open Sentences Involving Absolute Value)
Absolute value –
Solve an Absolute Value Equation
Ex: 1
Solve each open sentence.
1.) |𝑥 − 5| = 8
2.) |𝑥 + 9| = 2
3.) |𝑥 − 2 | = −5
4.) |2𝑥 − 3| = 17
Ex: 2
The temperature of an enclosure for a pet snake should be about 80 degree Fahrenheit, give or take
5 degrees. Solve |𝑥 − 80| = 5 to find the maximum and minimum of the temperatures.
Method 1 (Graphing)
Method 2 (Compound Sentence)
6 – 7: Graphing Inequalities in Two Variables
From the set (3,3),(0,2),(2,4),(1,0), which ordered pairs are part of the solution set for
y  2 x  4 ?

The graph of all of the ordered pairs fill a region on the coordinate plane called a
__________
____________ and an equation defines the ______________, or
edge. The inequalities ______ and ______ are represented by a dashed
boundary line and the inequalities ______ and _______ are represented by a
solid boundary line.
Example 1: y  2x 1

Steps for Graphing Inequalities:
1)
2)
3)
4)
Example 2: 2 y  4 x  6
Example 3: 3x  4 y  12

Example 4: y < 4
Example 5: x  1

6-8 Graphing Systems of Inequalities
System of Inequalities-
Ex: 1 (Solve by Graphing)
𝑦 < −𝑥 + 1
𝑦 ≤ 2𝑥 + 3
Ex: 2
𝑥 − 𝑦 < −1
𝑥−𝑦 >3