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MATH 1100
SECTION 3.7 Notes
Non-Linear Inequalities – Text Pages 187-194
THE SIGN OF A PRODUCT OR QUOTIENT
If a product or a quotient has an even number of negative factors, then
its value is positive.
If a product or quotient has an odd number of negative factors, then its
value is negative.
GUIDELINES FOR SOLVING NON-LINEAR INEQUALITIES
1. MOVE ALL TERMS TO ONE SIDE.
If necessary, rewrite the
inequality so that all non-zero terms appear on one side of the
inequality sign.
If the non-zero side of the inequality involves
quotients, bring them to a common denominator.
2. FACTOR. Factor the non-zero side of the inequality.
3. FIND THE INTERVALS. Use the factorization to find all solutions of
the equation corresponding to a given inequality. These numbers will
divide the real line into intervals. List the intervals determined by
these numbers.
4. MAKE A TABLE OR DIAGRAM. Use test values to make a table or
diagram of the signs of each factor on the interval. In the last row of
the table determine the sign of the product (or quotient) of these
factors.
5. SOLVE. Determine the solution of the inequality from the last row of
the sign table. Be sure to check whether the inequality is satisfied by
some or all of the endpoints of the intervals (this may happen if the
inequality involves  or  ).
Example 1: Solve and graph the following inequality:
x2  9
Step 1:
Step 2:
Step 3:
CORRESPONDING EQUATIONS:
Step 4:
Step 5:
Interval:
Graph:
Example 2: Solve and graph the following inequality:
x 2  5x  6  0
Step 1:
Step 2:
Step 3:
CORRESPONDING EQUATIONS:
Step 4:
Step 5:
Interval:
Graph:
Example 3: Solve and graph the following inequality:
x  2x  1x  3  0
Step 1:
Step 2:
Step 3:
CORRESPONDING EQUATIONS:
Step 4:
Step 5:
Interval:
Graph:
Example 4: Solve and graph the following inequality:
2x  6
0
x2
Step 1:
Step 2:
Step 3:
CORRESPONDING EQUATIONS:
Step 4:
Step 5:
Interval:
Graph:
Example 5: Solve and graph the following inequality:
x
 3x
x 1
Step 1:
Step 2:
Step 3:
CORRESPONDING EQUATIONS:
Step 4:
Step 5:
Interval:
Graph:
Example 6: Solve and graph the following inequality:
x  12  0
x  1x  2
Step 1:
Step 2:
Step 3:
CORRESPONDING EQUATIONS:
Step 4:
Step 5:
Interval:
Graph: