Solving Inequalities
Solving Inequalities by Using Addition and Subtraction

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 )

The solutions of an inequality can be graphed on a number line
or writtenusing setbuilder 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.

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 .

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} .
