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1.7 and 1.8
Graph an
Inequality’s
Solution Set
Set-builder
And Interval
Notation
LINEAR AND ABSOLUTE VALUE INEQUALITIES
Example 1
Graph the solution set of each.
a)
x3
b)
x  1
c)
1  x  3
There are two ways to represent solution sets of inequalities. They are
set-builder notation and interval notation. The table below shows
the nine possible types of intervals used to describe inequalities.
Interval
Notation
 2, 4
Set-Builder Notation
Graph
x  2  x  4
x  2  x  4
x  2  x  4
x  2  x  4
x x  2
x x  2
x x  4
x x  4
x x  
2, 4
2, 4
 2, 4
 2,  
2, 
 , 4
 , 4
 , 
The set of all real numbers
Example 2
a)
b)
c)
d)
Express the intervals in terms of inequalities and then graph.
State whether the interval is bounded or unbounded.
 1, 4
0, 9
 4, 
 , 5
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Properties of
Inequalities
The only thing to remember about solving inequalities is:
When multiplying or dividing by a negative number, we must reverse the
direction of the inequality.
Example 3
Solve. Graph the solution. Write the solution in set-builder
notation and interval notation.
a)
3x  5  7
b)
4  3x  7  2x
c)
 3  2x  1  3
Quadratic
Inequalities
One method of solving a quadratic inequality involves finding the solutions
of the corresponding equation, and then testing values in the intervals on a
number line determined by these solutions.
Steps
STEPS FOR SOLVING A QUADRATIC INEQUALITIES
Step 1 – Solve the corresponding quadratic equation.
Step 2 – Identify the intervals determined by the solutions of the equation.
Step 3 – Use a test value from each interval to determine which intervals form
the solution set.
Example 4
Solve the inequality x 2  x  12  0
Example 5
Solve the inequality
2
2 x 2  5 x  12  0
Rational
Inequalities
Inequalities involving rational expressions (fractions) are called rational
inequalities and are solved in a manner similar to the procedure for solving
quadratic inequalities.
Steps
STEPS FOR SOLVING A RATIONAL INEQUALITIES
Step 1 – Rewrite the inequality, if necessary, so that 0 is on one side and there
is a single fraction on the other side.
Step 2 – Determine the values that will cause either the numerator of the
denominator of the rational expression to equal 0. These values determine the
intervals on the number line to consider.
Step 3 – Use a test value from each interval to determine which intervals form
the solution set.
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1
Example 6 Solve
x4
Example 7
Solve
2x 1
3
x 3
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Solve Inequalities Absolute value means _______________ from 0 on a number line.
Involving Absolute Therefore, x  2 means “all the number 2 units away from 0.”
Value
Graphically, you can think of this:
To solve an absolute value inequality, we do the same process, but with
Inequalities rather than equalities.
x  2 means all the number less than 2 units from 0 on a number line.
Graphically, this is:
x  2 means all the number more than 2 units from 0 on the number line.
Graphically, this is:
Example 8
Solve and graph:
x4 3
Example 9
Solve and graph:
2x  3  5
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Extra Practice:
Solve each inequality. Write the solution set in interval notation.
1. 6 x   2 x  3  4 x  5
2.
2  4 x  5  x 1  6  x  2
4.
3  2x  7  13
x 1
5
2
6.
1  6x  5  4
7. x 2  x  6  0
8.
2 x 2  9 x  18  0
9. x  x  1  6
10.
x 2  4 x  4
3.
1
2
1
1
x  x   x  3 
3
5
2
10
5. 4 
11.
x 3
0
x5
12.
x 1
0
x4
13.
x6
 1
x2
14.
10
5
2x  3
15.
7
1

x2 x2
16.
4
3

x  2 x 1
Review:
17. Solve:
2x  3  x  3
18. Solve by completing the square:
x 2  6 x  12  0
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