The maximum distance the earth is from the sun is 152,000,000 km and the minimum distance is 147,000,000 km. Write an absolute value equation where d represents maximum and minimum distance between earth and the sun in millions of kilometers.

Question

The maximum distance the earth is from the sun is 152,000,000 km and the minimum distance is 147,000,000 km.  Write an absolute value equation where d represents maximum and minimum distance between earth and the sun in millions of kilometers.

ashleyisakitty · Answer

147,000 <=| d |<= 152,000 (i think)

anonymous · Answer

Didn't we do this yesterday?

mathstudent55 · Answer

$147000000 \le d \le 152000000 $

$-2 500 000 \le d -  149 500 000 \le 2 500 000 $

$ |d - 149500000 | \le 2500000 $

anonymous · Answer

yes..that's the answer i gave yesterday..
but it has to be in millions of km. So change 149500000 to 149.5 and 2500000 to 2.5

anonymous · Answer

How did you get 2.5 though?

mathstudent55 · Answer

Good point.

Final answer is:

$ |d - 149.5| \le 2.5 $

mathstudent55 · Answer

The difference between 152,000,000 and 147,000,000 is 5,000,000.
half of 5,000,000 is 2,500,000.

mathstudent55 · Answer

An absolute value inequality of the form 

$|X| \le k $

where X is an expression in x and k is a number is solved by changing it into the compound inequality

$ -k \le X \le k$

We started with the end result (I will now switch to millions of km to shorten the writing.):

$147 \le d \le 152 $

We need the two numbers in the beginning and end to be additive inverses to fit in with the compound inequality. To do so, we find the average of the numbers, (147 + 152)/2 = 149.5. Then we subtract that from all three "sides" of the compound inequality.

$147 - 149.5 \le d - 149.5 \le 152 - 149.5 $

which simplifies to

$-2.5 \le d - 149.5 \le 2.5 $

Now we have -2.5 and 2.5, the negative additives we need for the compound inequality, so we can turn it into an absolute value inequality:

$ |d - 149.5| \le 2.5 $