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# help algebra

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If a and b are integers such that a - b = 10, find the minimum value of a*b.

Jun 16, 2021

#1
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-5 * -5 = -25

I hope this helps

Jun 16, 2021
#3
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Hello, mnk!

-5 * -5 = 25      (nope)

`

a = 5       b = -5

a - b = 5 - (-5) = 10

a * b = 5 * -5 = -25   (yes)

Guest Jun 16, 2021
#2
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The term $a*b$ is nonnegative whenever both $a$ and $b$ have the same sign. It is negative whenever $a$ is positive and $b$ is negative.   Since $a$, $b$ are integers with $a = b+10$, these are all the possible ways that $ab$ can be negative:
\begin{eqnarray*}
a = 1, b = -9, ab = -9 \\
a = 2, b = -8, ab = -16 \\
a = 3, b = -7, ab = -21 \\
a = 4, b = -6, ab = -24 \\
a = 5, b = -5, ab = -25 \\
a = 6, b = -4, ab = -24 \\
a = 7, b = -3, ab = -21 \\
a = 8, b = -2, ab = -16 \\
a = 9, b = -1, ab = -9
\end{eqnarray*}

So the minimum value of $ab$ is $-25$.

Jun 16, 2021