To see whether the graph of an absolute valueinequality is sometimes, always, or never the union of two graphs, let's take the simplest absolute value expression.
|x|
Now, we will consider different inequalities involving it. Let's look at some examples.
First Inequality:&|x|< 3
Second Inequality:&|x|>3
Let's solve them!
First Inequality
To solve this inequality, we have to consider two cases, a non-negative and negative one.
Case 1:&x< 3
Case 2:&x> -3
We can tell that the first inequality is true when x is less than 3, and the second one is true when x is greater than -3. Therefore, we can graph the solution set on the number line as shown below.
As we can see, it is an intersection of two graphs.
Second Inequality
To solve this inequality, we also have to consider two cases, a non-negative and negative one.
Case 1:&x> 3
Case 2:&x< -3
We can tell that the first inequality is true when x is greater than 3, and the second one is true when x is less than -3. Therefore, we can graph the solution set on the number line as shown below.
As we can see, it is a union of two graphs. Therefore, we can tell that the graph of an absolute value inequality is sometimes the union of two graphs.
