1. 1.4
Solving Inequalities
OBJECTIVES
a Determine whether a given number is a solution of an
inequality.
b Graph an inequality on the number line.
c Solve inequalities using the addition principle.
d Solve inequalities using the multiplication principle.
e Solve inequalities using the addition principle and the
multiplication principle together.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 1
2. 1.4
Solving Inequalities
a
Determine whether a given number is a solution of an
inequality.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 2
3. 1.4
Solving Inequalities
SOLUTION
A replacement that makes an inequality true is called a
solution. The set of all solutions is called the solution
set. When we have found the set of all solutions of an
inequality, we say that we have solved the inequality.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 3
4. 1.4
a
Solving Inequalities
Determine whether a given number is a solution of an
inequality.
EXAMPLE
Determine whether 2 is a solution of x < 2.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 4
5. 1.4
Solving Inequalities
Determine whether a given number is a solution of an
a
inequality.
EXAMPLE
Determine whether 6 is a solution of
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 5
6. 1.4
b
Solving Inequalities
Graph an inequality on the number line.
A graph of an inequality is a drawing that represents its
solutions. An inequality in one variable can be graphed
on the number line. An inequality in two variables can be
graphed on the coordinate plane.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 6
7. 1.4
Solving Inequalities
b Graph an inequality on the number line.
EXAMPLE
The solutions are all those numbers less than 2. They are
shown on the number line by shading all points to the left
of 2. The open circle at 2 indicates that 2 is not part of the
graph.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 7
8. 1.4
Solving Inequalities
b Graph an inequality on the number line.
EXAMPLE
The solutions are shown on the number line by shading
the point for –3 and all points to the right of –3. The
closed circle at –3 indicates that –3 is part of the graph.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 8
9. 1.4
Solving Inequalities
b Graph an inequality on the number line.
EXAMPLE
The inequality is read “–3 is less than or equal to x and x is
less than 2,” or “x is greater than or equal to –3 and x is
less than 2.” In order to be a solution of this inequality, a
number must be a solution of both
and x < 2. We
can see from the graphs that the solution set consists of
the numbers that overlap in the two solution sets in
Examples 5 and 6.
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Slide 9
10. 1.4
Solving Inequalities
b Graph an inequality on the number line.
EXAMPLE
The open circle at 2 means that 2 is not part of the graph.
The closed circle at –3 means that is part of the graph. The
other solutions are shaded.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 10
11. 1.4
Solving Inequalities
c
Solve inequalities using the addition principle.
Any solution of one inequality is a solution of the
other—they are equivalent.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 11
12. 1.4
Solving Inequalities
THE ADDITION PRINCIPLE FOR INEQUALITIES
For any real numbers a, b, and c:
In other words, when we add or subtract the same
number on both sides of an inequality, the direction of
the inequality symbol is not changed.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 12
13. 1.4
Solving Inequalities
c
Solve inequalities using the addition principle.
As with equation solving, when solving inequalities, our
goal is to isolate the variable on one side. Then it is easier
to determine the solution set.
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Slide 13
14. 1.4
c
Solving Inequalities
Solve inequalities using the addition principle.
EXAMPLE
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Slide 14
15. 1.4
c
Solving Inequalities
Solve inequalities using the addition principle.
A shorter notation for sets is called set-builder notation.
is read
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Slide 15
16. 1.4
Solving Inequalities
THE MULTIPLICATION PRINCIPLE FOR INEQUALITIES
For any real numbers a and b, and any positive number c:
For any real numbers a and b, and any negative number c:
Similar statements hold for
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 16
17. 1.4
Solving Inequalities
THE MULTIPLICATION PRINCIPLE FOR INEQUALITIES
In other words, when we multiply or divide by a positive
number on both sides of an inequality, the direction of
the inequality symbol stays the same. When we
multiply or divide by a negative number on both sides
of an inequality, the direction of the inequality symbol is
reversed.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 17
18. 1.4
Solving Inequalities
d Solve inequalities using the multiplication principle.
EXAMPLE
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Slide 18
19. 1.4
Solving Inequalities
d Solve inequalities using the multiplication principle.
EXAMPLE
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Slide 19
20. 1.4
Solving Inequalities
e
Solve inequalities using the addition principle and the
multiplication principle together.
Remember to reverse the inequality symbol when
multiplying or dividing on both sides by a negative
number.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 20
21. 1.4
e
Solving Inequalities
Solve inequalities using the addition principle and the
multiplication principle together.
EXAMPLE
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Slide 21
22. 1.4
Solving Inequalities
Solve inequalities using the addition principle and the
e
multiplication principle together.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 22
23. 1.4
e
Solving Inequalities
Solve inequalities using the addition principle and the
multiplication principle together.
EXAMPLE
First, we use the distributive law to remove parentheses.
Next, we collect like terms and then use the addition and
multiplication principles for inequalities to get an
equivalent inequality with x alone on one side.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 23
24. 1.4
Solving Inequalities
Solve inequalities using the addition principle and the
e
multiplication principle together.
EXAMPLE
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Slide 24
25. 1.4
Solving Inequalities
Solve inequalities using the addition principle and the
e
multiplication principle together.
EXAMPLE
The greatest number of decimal places in any one
number is two. Multiplying by 100, which has two 0’s,
will clear decimals. Then we proceed as before.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 25
26. 1.4
Solving Inequalities
Solve inequalities using the addition principle and the
e
multiplication principle together.
EXAMPLE
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Slide 26
27. 1.4
Solving Inequalities
Solve inequalities using the addition principle and the
e
multiplication principle together.
EXAMPLE
The number 6 is the least common multiple of all the
denominators. Thus we first multiply by 6 on both sides to
clear the fractions.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 27
28. 1.4
Solving Inequalities
Solve inequalities using the addition principle and the
e
multiplication principle together.
EXAMPLE
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Slide 28
29. 1.4
Solving Inequalities
Solve inequalities using the addition principle and the
e
multiplication principle together.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 29
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