The absolute value of the difference between the acceptable number of words and 500 words cannot be greater than 30.
Inequality: |w-500|≤30 Words: 470 to 530 words
Practice makes perfect
We have been told that for an essay contest entries can have 500 words with an absolute deviation of at most 30 words. Let w be the number of words written. The absolute deviation is the difference between w and 500. We can state this as an inequality.
|w- 500|≤ 30
Now, we will solve this absolute value inequality to find the acceptable numbers of words. To do this, we will create a compound inequality by removing the absolute value. In this case, the solution set can be written as the following.
Absolute Value: |w-500| &≤ 30
Compound: - 30 ≤ w-500 &≤ 30
This compound inequality means that the distance from w-500 is greater than or equal to - 30 andless than or equal to 30.
w-500≥- 30 and w-500≤ 30
Let's isolate w in both of these cases to find the solution set.
Case 1
We can solve the inequality by performing inverse operations on both sides of the inequality. Let's solve the first case.
This inequality tells us that all values less than or equal to 530 will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the overlap of the solution sets. Let's recombine our cases back into one compound inequality.
First Solution Set:& 470≤ w
Second Solution Set:& w≤ 530
Intersecting Solution Set:& 470≤ w≤ 530
The acceptable number of words that can be written in the essay is between 470 and 530.
