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.
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.
Compound inequalities can be split up or solved in one step like the following examples. Note that all inequality symbols face the same direction when combined.
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?
