Showing posts with label interval notation. Show all posts
Showing posts with label interval notation. Show all posts

Solving Absolute Value Equations, Quadratic (Polynomial), and Rational Inequalities using Sign Charts

Free math notes on solving absolute value equations and polynomial (including quadratic) and rational inequalities using a sign chart.
   1.  Absolute Value Equations
   2.  Absolute Value Inequalities
   3.  Quadratic Inequalities
   4.  Polynomial and Rational Inequalities

YouTube Videos: Click on a problem to see it worked out in YouTube.

Absolute Value Equations:

Absolute Value Inequalities

Quadratic Inequalities:

Polynomial and Rational Inequalities

Absolute Value Inequalities

Absolute value inequalities and equations are a bit tricky to work with.  There are basically three cases or situations that can arise when working with them.  By guessing and checking we can answer the following three questions.
Tip: We can easily generalize the above result so that we can use this idea as a template when solving equations and inequalities with absolute values in them. ( Assume n > 0 )

   Case 1: | | = n can be solved using X = -n or X = n.
   Case 2: | | < n can be solved using -n < X < n.
   Case 3: | | > n can be solved using X < -n or X > n.

Use the following steps to solve an absolute value equation or inequality.

   Step 1: Isolate the absolute value.
   Step 2: Identify the case and apply the appropriate theorem.
   Step 3: Solve the resulting equation or inequality.
   Step 4: Graph the solution set and express it in interval notation.

Instructional Video: Absolute Value Inequalities

Solve and graph the solution set.
In the three cases listed above notice that we took care to say that n > 0.  The next three problems illustrate some of the situations encountered when n = 0. Plug in some numbers and see what happens.
YouTube Videos:

Linear Inequalities (one variable)

Solve and graph the solution set. 
(I.N. stands for interval notation here.)
Instructional Video: Introduction to Interval Notation
Compound inequalities can be split up or solved in one step like the following examples.

Instructional Video: Compound Inequalities

Average Problem: Clint wishes to earn a B which is an average of at least 80 but not more than 90.  What range must he score on the fourth exam if the first three were 65, 75, and 90?
Commission Problem: Bill earns $12.00 plus $0.25 for every person he gets to register to vote. How many people must he register to earn at least $50.00 for the day?
Video Examples on YouTube: