Topics:
• Quick review of the first method of solving absolute value inequalities, expanded to also look at the graph. See also Example 12.4(d) in the textbook.
• Solving linear inequalities: this is just like solving linear equations, except that when you divide or multiply both sides by a negative number, you must reverse the direction of the inequality. This may result in a double inequality being put in the wrong order, and then you have to fix that up: see Example 12.1(c).
• Solving polynomial inequalities of degree greater than 1: the method is basically the same as the first method we used for absolute value inequalities, except that we want to make one side 0 first, in order to make it easier to solve the related equation. See Examples 12.2
• Solving rational inequalities: the method is basically the same as the method for solving polynomial inequalities, except that we also need to consider the zeroes of the denominator along with the solutions of the related equation when we cut the real line up into intervals. (However, zeroes of the denominator will never be part of the solution set!) See the example on the handout: also Examples 12.4(a-c)
The methods for solving polynomial inequalities (of degree >1) and rational inequalities are outlined below the fold, at the bottom of this post.
Homework:
• Reread and review the examples worked in class. It is also very useful to study the other parts of Examples 12.1, 12.2, and 12.4.
• Do the assigned problems from Session 12
• Do (if you have not already done it) Exercise 11.4, especially 11.4(d). I will call someone at random to put this on the board.
• Do the WeBWorK: start early! Due by Tuesday (tomorrow!) 11 PM
• No Warm-Up this time.
• Make sure you check the Checklist!
Methods for solving polynomial and rational inequalities outlined below the fold…
Method for solving polynomial inequalities (degree > 1):
• Make one side of the inequality 0, if necessary
• Solve the related equation: use all the tools from Session 10 to find all the real roots of the polynomial. If you guess roots by looking at the graph, make sure that you check that they really are roots, either by substitution or by synthetic division.
• The solutions of the related equation cut the real line into a number of intervals. Either
•• Test a point in the interior of each interval, to see if the inequality is true there
– OR –
•• Look at the graph of the polynomial to see where its graph lies above or below the x-axis, as required
It is not a bad idea to do both, as a check.
• Finally, consider whether or not the endpoints are included in the solution set, and write your solution set in the form asked for in the problem.
The method for rational inequalities is basically the same, with changes as follows:
• Make one side of the inequality 0, if necessary
• Combine all terms on the other side into a single rational expression, if there is more than one term.
• Solve the related equation: use all the tools from Session 10 to find all the real roots of the polynomial in the numerator. If you guess roots by looking at the graph, make sure that you check that they really are roots, either by substitution or by synthetic division.
• Also find all zeroes of the denominator of the rational expression: these will represent either places where there are vertical asymptotes, or places where there are holes in the graph. Again, use all the tools of Session 10 to find these zeroes.
• The solutions of the related equation, along with the zeroes of the denominator, cut the real line into a number of intervals. Either
•• Test a point in the interior of each interval, to see if the inequality is true there
– OR –
•• Look at the graph of the polynomial to see where its graph lies above or below the x-axis, as required
It is not a bad idea to do both, as a check.
• Finally, consider whether or not the endpoints are included in the solution set, and write your solution set in the form asked for in the problem. Remember that the zeroes of the denominator will never be in the solution set, because they are not even in the domain of the rational function!
