Solving Inequalities
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Solving linear inequalities is the same
as solving linear equations...
with one very important exception...
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when you multiply or divide an inequality by a negative value, it
changes the direction of the inequality.
Consider:
Look at this true statement:
Suppose we multiply both sides by -1.
What is the relationship between these two numbers ?
5>3
(-1)(5) ? (3)(-1)
-5 ? -3
ANS: -5 is less than -3 because it is further to the left
on the number line.
-5 < 3
So, we must change the direction of the inequality when we multiply (or divide) by a
negative number in order to get the correct answer.
Inequality Notation:
SYMBOL
MEANING
less than
greater than
less than or equal to
greater than or equal to
Solve and graph the solution set of: 2x - 6 < 2
Add 6 to both sides.
Divide both sides by 2.
Open circle at 4 (since x can not equal 4) and
an arrow to the left (because we want values
less than 4).
2x - 6 < 2
2x < 8
x<4
Solve and graph the solution set of: 5 - 4x 13
Subtract 5 from both sides.
Subtract x from both sides.
Divide both sides by -4, and don't forget to
change the direction of the inequality !
(We divided by a negative.)
Closed circle at -2 (since x can equal -2) and an
arrow to the right (because we want values larger
than -2).
5 - 4x 13
-4x 8
x -2