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:

Linear Inequalities (Two Variables)

When graphing an equation like y = 3x − 6 we know that it will be a line.  The graph of a linear inequality such as y >= 3x − 6, on the other hand, gives us a region of ordered pair solutions.

     

Not only do the points on the line satisfy this linear inequality - so does any point in the region that we have shaded.  This line is the boundary that separates the plane into two halves - one containing all the solutions and one that does not. Therefore, from the above graph, both (0, 0) and (−2, 4) should solve the inequality.

  

Use a test point not on the boundary to determine which side of the line to shade when graphing solutions to a linear inequality.  Usually the origin is the easiest point to test as long as it is not a point on the boundary.

Instructional Video: Linear Inequalities in Two Variables

Graph the solution set.

 

If the test point yields a true statement shade the region that contains it.  If the test point yields a false statement shade the opposite side.

When graphing strict inequalities, inequalities without the equal, the points on the line will not satisfy the inequality; hence, we will use a dotted line to indicate this.  Otherwise, the steps are the same.

Graph the solution set.

Given the graph determine the missing inequality.

 

Video Examples on YouTube: