# Sum of Absolute Values

I was going over a question and I wanted your opinion(s) on it:

The product of two numbers is 6 and one of the numbers is 5 less than the other. What is the absolute value of the sum of the two numbers?

The numbers I got are 1 and -6 because $b(b+5)=6$. Now the sum of absolute values would be $|1-6|$, right, so 5? According to the book the answer is 7. Apparently the book expects me to do $|1| + |-6| = 7$. Am I right and the answer in the book is wrong?

Edit: It seems my numbers are wrong, here is how I got them. Please correct me if I am wrong.

$$(b+5)(b) = 6$$

so I get $$b^2 + 5b -6 = 0$$

$b= 1$ and $b=-6$ by solving the quadratic equation.

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If the book is asking for the sum of the absolute values, as your title suggests, then 7 is the correct answer. If the book is asking for the absolute value of the sum, then 5 is correct. – Old John Jun 25 '12 at 9:28
The numbers are wrong: $1\cdot(-6)=-6\neq6$. Also, the difference between $1$ and $-6$ is $7$, not $5$. – Jyrki Lahtonen Jun 25 '12 at 9:31
So they are - I assumed he had at least done that bit right! – Old John Jun 25 '12 at 9:32
Sorry i was suppose to enter those values back in the equation. – Rajeshwar Jun 25 '12 at 9:40
Your solution of the quadratic is correct, which means that the smaller of the two numbers is either -6 or 1. In the first case, the two numbers would be -6 and -1, whereas in the second case, the numbers would be 1 and 6. In either case, the sum of the absolute values and the absolute value of the sum are both equal to 7. – Old John Jun 25 '12 at 9:43

Let $x$ be the smaller of the numbers, so the other is $x+5$. We have the equation $$x(x+5)=6.$$ Solve for $x$ and check that $|x+(x+5)|=7$ no matter which solution of the quadratic you use as $x$.