Download Lecture notes for Section 6.5

Int. Alg. Notes
Section 6.5
Page 1 of 4
Section 6.5: Rational Inequalities
Big Idea: Rational inequalities can be solved by simplifying the inequality to a single rational expression on
one side of the inequality and zero on the other side.
Big Skill: You should be able to solve a rational inequality by moving all terms onto one side and combining
them into a single rational expression, then identifying where all the zeros of all the factors are, then testing
ranges between those zeros to find intervals that satisfy the inequality.
Steps for Solving a Rational Inequality
1. Write the inequality so it is a single rational expression on one side of the inequality and zero on the
other side. Completely factor the numerator and denominator of the rational expression.
2. Determine all numbers that make the factors of the rational expression zero.
3. Use those zeros to separate the real number line into intervals. We do this because the factor will be
positive on one side of the zero and negative on the other side of the zero.
4. Choose a test point in each interval, and determine the sign of the rational expression at that test point.
If the sign for a test value matches the inequality, then the entire interval containing the test point is a
solution.
Practice:
1. Solve
2x  3
 0 , and graph the solution set.
x2
Interval
Test Point
Sign of factor:
Sign of factor:
Sign of factor:
Sign of factor:
Sign of rational
expression
Conclusion
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
2. Solve
Section 6.5
Page 2 of 4
2x  3
 1 , and graph the solution set.
x2
Interval
Test Point
Sign of factor:
Sign of factor:
Sign of factor:
Sign of factor:
Sign of rational
expression
Conclusion
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
3. Solve
Section 6.5
Page 3 of 4
1
3

, and graph the solution set.
x  4 2x 1
Interval
Test Point
Sign of factor:
Sign of factor:
Sign of factor:
Sign of factor:
Sign of rational
expression
Conclusion
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
4. Solve
Section 6.5
Page 4 of 4
1
5
 2
, and graph the solution set.
x  2x  3 x 1
2
Interval
Test Point
Sign of factor:
Sign of factor:
Sign of factor:
Sign of factor:
Sign of rational
expression
Conclusion
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.