Linear Equations And Inequalities In One Variable Worksheet With Answer Key Page 4
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2
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8(2-59)
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2.9 Solving Inequalities and Applications
E X A M P L E
3
Reversing the inequality symbol
Solve and graph the inequality 5
5x
1
2(5
x ).
Solution
5
5x
1
2(5
x)
Original inequality
5
5x
11
2x
Simplify the right side.
5
3x
11
Add 2x to each side.
3x
6
Subtract 5 from each side.
x
2
Divide each side by
3, and reverse the inequality.
– 6 – 5 – 4 – 3
– 2 – 1
0
1
The inequalities 5
5x
1
2(5
x) and x
2 have the same graph, which
F I G U R E 2 . 1 4
is shown in Fig. 2.14.
We can use the rules for solving inequalities on the compound inequalities that
we studied in Section 2.8.
E X A M P L E
4
Solving a compound inequality
2
x
Solve and graph the inequality
9
7
5.
3
Solution
2
x
9
7
5
Original inequality
3
2
x
9
7
7
7
5
7
Add 7 to each part.
3
2
x
2
12
Simplify.
3
3
3
2
x
3
3
( 2)
12
Multiply each part by
.
2
2
2
3
2
3
x
18
Simplify.
– 3
0
3
6
9 12 15 18
Any number that satisfies
3
x
18 also satisfies the original inequality.
F I G U R E 2 . 1 5
Figure 2.15 shows all of the solutions to the original inequality.
There are many negative numbers in Example 4, but the
C A U T I O N
inequality was not reversed, since we did not multiply or divide by a negative num-
ber. An inequality is reversed only if you multiply or divide by a negative number.
E X A M P L E
5
Reversing inequality symbols in a compound inequality
Solve and graph the inequality
3
5
x
5.
Solution
3
5
x
5
Original inequality
3
5
5
x
5
5
5
Subtract 5 from each part.
8
x
0
Simplify.
( 1)( 8)
( 1)( x )
( 1)(0)
Multiply each part by
1, reversing
the inequality symbols.
8
x
0
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