Absolute-value inequality word problem

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My son has the following problem in his homework. The answer is the in the back of the book and I can't make any sense of how they got it. Help!

Problem: On your basketball team, the starting players' scoring averages are between 8 and 22 points per game. Write an absolute-value inequality describing the scoring averages for the players.

Answer: l x-15 l is less than or equal to 7.

Thanks! Cathy
 
toneymom said:
On your basketball team, the starting players' scoring averages are from 8 through 22 points per game. Write an absolute-value inequality describing the scoring averages for the players.
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If we remove the absolute value symbols from the following inequality

|expression| < value

then we get the following compound inequality

-value < expression < value

This is a basic property of absolute value inequalities that needs to be memorized. (If one understands absolute value to be a distance from zero on the Real number line, then this property is self-explanatory.)

Therefore, the inequality

|x - 15| <= 7

becomes the following compound inequality, once we use the property above to remove the absolute value symbols.

-7 <= x - 15 <= 7

Well, this process is reversible. I mean, once we come up with the numbers 15 and -7 and 7, and we write

-7 <= x - 15 <= 7

it's simply a matter of rewriting it using absolute value form

|x - 15| <= 7

So, now the question becomes: where did the -7, 7, and 15 come from?

Let x be a particular starter's scoring average.

Given: 8 <= x <= 22

The average of the endpoints of this given range of scoring averages is 15. I mean, the average of the lowest scoring average (8) and the highest scoring average (22).

Subtracting this average from each part in the given compound inequality above gets us to

-7 <= x - 15 <= 7

I'm sure that your son won't care about the following, but the description of the scoring averages given by the answer in this exercise is that the difference between a particular player's own scoring average (x) and the average of the starters' high-and-low averages (15) cannot be smaller than -7 (when the particular player's scoring average is less than the high-and-low average) or exceed 7 (when the particular player's scoring average is more than the high-and-low average).

In other words, a starter's scoring average cannot be more than 7 points above or below the average (of the group's high-and-low scoring averages) of 15.

 
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