2. Important Idea
In previous sections, we
have been solving equalities,
or equations. Now we are
going to solve inequalities.
The methods of solving
equalities and inequalities
are similar but there are
important differences.
11. Important Idea
Principles for solving
inequalities:
1. Add or subtract the
same number on both
sides of the inequality.
12. Important Idea
Principles for solving
inequalities:
2. Multiply or divide both
sides of the inequality by
the same positive number.
13. Important Idea
Principles for solving
inequalities:
3. Multiply or divide both
sides of the inequality by
the same negative number
and reverse the direction of
the inequality.
14. Example
2 3 5 2 11x x≤ + < +
Solve. Write your answer
using interval notation.
15. Try This
5 2 1 7x x− ≤ − ≤ +
[ ]2,8−
Solve. Write your answer
using interval notation.
16. Example
4 3 5 18x< − <
Solve. Write your answer
using interval notation.
Graph your answer on a
number line.
17. Try This
Solve. Write your answer
using interval notation.
Graph your answer on a
number line.
2
,2
3
− ÷
2 4 3 6x− < − <
18. Important Idea
The solutions of the form
( ) ( )f x g x< consist of
intervals on the x axis where
the graph of f is below the
graph of g.
23. Important Idea
Solving an inequality
depends only on knowing
the zeros of the function
and where the graph is
above or below the x-
axis. The zeros are where
the function touches the x
axis.
27. Example
Find the exact solutions:
Confirm with your calculator
6
( 5)( 2) ( 8) 0x x x+ − − ≤
28. Important Idea
Steps for solving inequalities:
1. Write the inequality in one
of these forms:
( ) 0f x > ( ) 0f x ≥
( ) 0f x < ( ) 0f x ≤
29. Important Idea
Steps for solving inequalities:
2. Determine the zeros of f,
exactly if possible,
approximately otherwise.
30. Important Idea
Steps for solving inequalities:
3. Determine the intervals on
the x axis where the graph is
above or below the x axis.
31. Example
A store has determined the
cost C of ordering and storing
x laser printers.
300,000
2C x
x
= +
The delivery truck can bring
at most 450 printers. How
many should be ordered to
keep the cost below $1600?