February 8th
February 8th
Solving Inequalities
Solving Inequalities by Using Addition and Subtraction
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For all numbers a, b, and c, the
following are true. |
| Addition and Subtraction
Properties of Inequalities |
- If a > b, then a + c > b + c and a - c >
b - c. (Also true for
 )
- If a < b, then a + c < b + c and a - c <
b - c. (Also true for
 )
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The solutions of an inequality can be graphed on a number line
or writtenusing set-builder notation
Example
Solve 3 m - 7 > 4 m + 1. Check your solution, and graph it
on a number line.
Solution
3m - 7 > 4m + 1
3m - 7 - 3m > 4m + 1 - 3m
- 7 > m + 1
- 7 - 1 > m + 1 - 1
- 8 > m or m < -8
In set builder notation, the solution set is {m | m < -8,
which is read "the set of all numbers m such that m is less
than -8". Only numbers less than - 8 substituted into the
original inequality should yield a true statement.
Since only the number less than - 8 yields a true statement,
the solution checks.Graph the point - 8 using an open circle,
since - 8 is not part of the solution.Then draw a heavy arrow to
the left to indicate numbers less than - 8.
Solving Inequalities by Using Multiplication and Division
When you multiply or divide each side of an inequality by a
negative number, you must reverse the direction of the inequality
symbol.
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For all numbers a, b, and c, the
following are true. |
| Multiplication and Division
Properties for Inequalities |
- If c is positive and a < b, then ac < bc
and
 , and if c is positive
and a b, then ac > bc and  .
- If c is negative and a < b, then ac > bc
and , and if c is negative and
a > b, then ac < bc and .
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These properties also hold true for inequalities involving and
.
Example
Solve -5 y 12 and check your solution.
Solution
-5 y 12
 Divide each side by -5 and change the
to .
Check: Let y be 2.4 and any number
greater than -2.4, such as 0.
| -5(-2.4) 12 |
-5(0) 12 |
12 12  |
0 12 |
In set builder notation, the solution set is { y | y
2.4} .
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