Showing posts with label inequalities. Show all posts
Showing posts with label inequalities. Show all posts

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:

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.

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:

Systems of Linear Inequalities (Two Variables)

The systems of linear inequalities that we will be solving consist of two linear inequalities and two variables.  To solve these we will graph the solution sets of both linear inequalities and then determine where the sets intersect.  Any point in the overlap of the graphs will be a solution to the system.

Instructional Video: Systems of Linear Inequalities

Graph the solution set:
The above solution suggests that (−5, 3) is a solution because it is shaded.  Check it and others to see if it solves both inequalities.

These graphs can sometimes get messy so do your best to think about the solution before actually shading.  Use pencil and a good eraser when working on these problems.

Given the graph, determine the system.

Graph the systems of inequalities.
Graph the system:

Video Examples on YouTube: