Question 1122951: A poll found that a particular group of people read an average of 12.3 books per year. The pollsters are 99% confident that the result from this poll is off by fewer than 2.74 books from the actual average x. Express this situation as an inequality involving absolute value, and solve the inequality for x to determine the interval in which the average is likely to fall.Express the situation as an inequality involving absolute value
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the average is 12.3 booke per year.
the pollsters are 99% confident that the actual results will not be off by more than 2.74 books plus or minus.
that means the actual results can be as little as 12.3 minus 2.74 or as much as 12.3 plus 2.74.
if x is the amount that the actual results can be off, then you get:
x - 12.3 < 2.75 or x - 12.3 > -2.75
this can be written as -2.75 < x - 12.3 < 2.75
this leads to the absolute value expression that says |x - 12.3| < 2.75
this absolute value expression is interpreted as:
when (x - 12.3) is positive, the expression becomes (x - 12.3) < 2.75
when (x - 12.3) is negative, the expression becomes (x - 12.3) > - 2.75
when you combine these two expressions together, you get -2.75 < (x - 12.3) < 2.75.
for example, let's' assume that the actual value is 2.5 units off the average.
then the actual value will be between 12.3 - 2.5 and 12.3 + 2.5.
this makes the actual value between 9.8 and 14.8.
if the actual value is 14.8, then the expression |14.8 - 12.3| < 2.75 becomes |2.5| < 2.75, which is true.
if the actual value is 9.8, then the expression |9.8 - 12.3| < 2.75 becomes |-2.5| < 2.75, which is also true, since |-2.5| is equal to 2.5 and that is smaller than 2.75.
here's a reference on absolute value expressions you might find helpful.
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note that the strict definition of absolute value expressions is:
if A is positive, then |A| = A.
if A is negative, then |A| = -A.
this leads to:
if |A| < B, and A is positive, then A < B.
if |A| < B, and A is negative, then -A < B.
in the expression -A < B, if you multiply both sides of the expression by -1, you get A > -B.
that leads to:
if |A| < B, and A is positive, then A < B.
if |A| < B, and A is negative, then A > -B.
both A and B represent expressions.
in your example of:
|x - 12.3| < 2.75
(x - 12.3) would be the expression shown as A.
2.75 would be the expression shown as B.
here's some references on absolute value expressions you might find helpful.
https://www.basic-mathematics.com/definition-of-absolute-value.html
https://www.purplemath.com/modules/absineq.htm
http://www.umsl.edu/~defreeseca/intalg/ch2extra/absineq.htm
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