How Far Away?

How Far Away?


Consider the following situations.

Situation 1: You are walking up and down the block in front of your school. You notice that if you are more than 100 yards from the front of school, you lose your internet connection.
Situation 2: You are walking up and down the block in front of your school. You notice that if you are fewer than 50 yards from the front of school, you can hear the band practicing.

How are these two situations the same? How are they different?

Absolute value can be used to describe the distance from the front of school. We will look at ways to solve absolute value inequalities by first examining a graph.

 

Solving Absolute Value Inequalities

Solving Absolute Value Inequalities


The absolute value of a number represents the distance that number would be from zero on a number line. This function can be useful in representing situations like the one described earlier. When solving an absolute value equation, we look for the one or two values, or possibly none, that will satisfy the given equation. For example, |x - 3| + 1 = 4 could be solved in the following manner:

|x - 3| + 1 = 4  
|x - 3| + 1 - 1 = 4 - 1
|x - 3| = 3
Subtract 1 from both sides to isolate the absolute value.
x - 3 = 3 or x - 3 = -3 Write two equations to solve: one to solve if what is inside the absolute value sign is equal to 3, the other if what is inside the absolute value sign is equal to -3.
x = 6 or x = 0 Find the two solutions.

Consider the question: When would an absolute value equation only have one solution?

Now that we have thought briefly about solving an absolute value equation, let's look at solving an absolute value inequality.

Absolute Value Activity

 

Practice Solving Absolute Value Inequalities

Practice Solving Absolute Value Inequalities


Use the interactive graph to match the inequality to its solution.

Absolute Value Activity