Remember that absolute values tell us the distance away from something. In an inequality, absolute values tell us if the distance is greater than or less than a given number.
See solution.
Practice makes perfect
Before we can solve either of these inequalities, we need to remember that absolute values tell us the distance away from something. In an inequality, absolute values tell us if the distance is greater than or less than a given number. Let's look at how we solve each one and then compare.
First Inequality
Let's think about what this inequality is telling us.
|w-9|≤2
This can be translated to a verbal phrase.
The distance from w-9 must be less than or equal to 2 units away.
If we want the distance away from something to be less than some value, it means we want it to stay close to the center. We want to contain our solutions in one central set. This is an andcompound inequality which we can write as the following.
-2≤ w-9≤ 2
We can solve this by adding 9 to all three parts of the compound inequality.
7≤ w≤ 11
Second Inequality
Once again, let's think about what the inequality is telling us.
|w-9|≥2
This can be translated to a verbal phrase.
The distance from w-9 must be greater than or equal to 2 units away.
If we want the distance away from something to be greater than some value, it means we want it to be away from the center. We want to separate our solutions into two disjointed sets. This is then an or compound inequality.
-2≥ w-9 or 2≤ w-9
We solve each of these inequalities separately.
7≥ w or 11≤ w
The Difference
When solving absolute value inequalities, you really need to think about the meaning of the absolute value in terms of distance. An inequality where the distance should be greater than the given value should be an or compound inequality. An inequality where the distance should be less than the given value should be an "and" compound inequality.
