Showing posts with label Absolute value. Show all posts
Showing posts with label Absolute value. 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.
Notes:
   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












Simplifying Radical Expressions

Now when working with square roots and variables we should be a bit careful.  The variable could represent a positive or negative number so we must ensure that it is positive by making use of the absolute value.
To avoid this technicality many textbooks state, at this point, that we assume all variables are positive. If not, use the absolute values as in the following problems.
Simplify. (Assume variables could be negative.)
To avoid many of the technicalities when working with nth roots we will assume, from this point on, that all the variables are positive.

Simplify. (Assume all variables represent positive numbers.)
Students often try to simplify the previous problem. Be sure to understand what makes the following problems all different.
       
The property:
says that we can simplify when the operation is multiplication.  There is no corresponding property for addition or subtraction.

Video Examples on YouTube:






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: