What happens if both y and x are between absolute values signs?
Example absolute value inequality: |y| < |x| . See solution for details.
Practice makes perfect
In previous exercises we have seen that an absolute value expression of y=|x| gives us a vertical V-shaped graph. A graph of the form |y|=x gives us a horizontal V-shaped graph. We can combine these cases and explore the relation |y|=|x|. This has two possibilities.
If y > 0, then we obtain the relation y = |x|.
If y < 0, then we obtain the relation y = - |x|.
Therefore, its graph would look like shown below.
We now have four regions. We can set an inequality whose solution set corresponds to two of these ones. Let's try, for example, |y| < |x|. To find the regions we would need to shade we can use a test point from each region. Let's use (0,1), (-1,0), (0,-1), and (1,0).
Let's do an example. We will try substituting (0,1).
We can see that the region where this point lies is not part of the solution set, as substituting it in the inequality does not give us a true statement. We can do the same with the other test points and find the solution set.
Test point
Evaluate |y| < |x|
Simplify
Is it a solution ?
( 0, 1)
| 1| ? <| 0|
1 ≮ 0
*
( - 1, 0)
| 0| ? <| -1 |
0 ≤ 1
✓
( 0, -1)
| - 1| ? <| 0|
1 ≮ 0
*
( 1, 0)
| 0| ? <| 1 |
0 ≤ 1
✓
Using these conclusions, we can now shade the corresponding areas of the solution set.
Notice that this is not the only possibility. We could have used this same inequality relation after applying a dilation or a translation. In general, any time we have both y and x between absolute value signs we will have a similar situation.
