Let d dollars be the initial amount in your bank account.
We have been given that you spent $22, so the amount left after spending $22 will be: .
We are also told that after spending $22, you have at least $28. This means that the amount left after spending $22 will be greater than or equal to $28.
We can represent this information in an equation as:
Therefore, the inequality represents the initial amount of money you had.
Now let us solve for d by adding 22 to both sides of our inequality.
Therefore, initially you had at least $50.
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Then in the entire integrand, set , so that . The integral is then equivalent to
Note that by letting , we are enforcing an invertible substitution which would make it so that requires or . However, is positive over this first interval and negative over the second, so we can't ignore the absolute value.
So let's just assume the integral is being taken over a domain on which so that . This allows us to write
We can show pretty easily that
which means the integral above becomes
Back-substituting to get this in terms of is a bit of a nightmare, but you'll find that, since , we get