Absolute Value Inequalities

Absolute value inequalities and equations are a bit tricky to work with.

Absolute Value Equations and Inequalities Playlist

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

Introduction to Inequalities and Interval Notation

All of the steps that we have learned for solving linear equations are the same for solving linear inequalities except one.  We may add or subtract any real number to both sides of an inequality and we may multiply or divide both sides by any positive real number.

Solving Linear Inequalities Playlist on YouTube

The only new rule comes from multiplying or dividing by a negative number.

So whenever we divide or multiply by a negative number we must reverse the inequality. It is easy to forget to do this so take special care to watch out for negative coefficients.

Notice that we obtain infinitely many solutions for these linear inequalities.  Because of this we have to present our solution set in some way other than a big list.  The two most common ways to express solutions to an inequality are by graphing them on a number line and interval notation.

Note: We use the following symbol to denote infinity:

Tip: Always use round parentheses and open dots for inequalities without the equal and always use square brackets and closed dots for inequalities with the equal.

Video Examples on YouTube:

Intermediate Algebra Exam #1

Click on the 10 question exam covering topics in chapters 1 and 2. Give yourself one hour to try all of the problems and then come back and check your answers.

[ IA Sample Exam #1 ]

 

 

 

   

Solve and graph the solution set. In addition, express the solution set using interval notation.

Sketch the graph and give the domain and range.

  

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