My teacher says that the following equation:$$-9-1(9x-6) = 3(4x+6)$$Should be simplified at the first stage to:$$-9-9x+6 = 12x+18$$He says that on the left side of the equation, we should multiply $-1$ by $9$ and $-6$ respectively, but to me this reads slightly different.
I interpret this as "negative $9$ minus $1$ times $9$ minus $1$ times $6$" which would simplify to:$$-9-9x-6$$The difference being that in my simplification, $6$ is subtracted from $-9-9x$ whereas in his simplification, $6$ is added to $-9-9x$.
I assume that he is right (he is the teacher after all) but I don't quite understand why I am wrong. I think this all comes down to the "minus $6$" vs. "negative $6$" issue. If you read the left side of the initial equation:$$-9-1(9x-6)$$If the '$-1$' part is to be read as "negative $1$" rather than "minus $1$" then what is the operation in between '$-9$' and '$-1$'. Is it a multiplication?
I often get confused by minus vs negative, does anyone have any tips or tricks for understanding this situation a bit better.
$\endgroup$ 33 Answers
$\begingroup$welcome to stack exchange.
The difference between minus and negative can be a bit subtle. It basically comes down to the following: $x-y=x+(-y)$. On the lefthand side we are considering "$x$ minus $y$", on the righthand side we are considering "$x$ plus negative $y$". Technically, the "minus" notation on the left side is just shorthand for the "plus negative" notation on the right side. In your example we are therefore looking at$$-9-1(9x-6)=-9+(-1)(9x+(-6))=-9+(-1)\cdot 9x+(-1)(-6)=-9x-3.$$If you want you can fill in some values of $x=1,2,3,-1,-2,-3$, work out the brackets like normal and you will see that the end results should be the same.
$\endgroup$ $\begingroup$In your way it must be:$$-9-1(9x-6)=-9-[1(9x-6)]=-9-[9x-6]=-9-9x+6$$because you want to multiply first, then subtract (minus).
$\endgroup$ $\begingroup$Will it convince you to try with some actual numbers?
If, for example, we set $x=2$, then we have$$ -9 -1(9x-6) = -9 -1(9\cdot 2 - 6)$$The value inside the parenthesis is $9\cdot 2-6 = 18-6 = 12$, so the value of the whole expression is$$ -9 - 1\cdot 12 = -9 - 12 = -21 $$
With your expression, you get$$ -9 -9x - 6 = -9 - 18 - 6 = -33 $$Since $-33$ is not the same as $-21$, your rewriting has not preserved the value of the expression when $x=2$. (In fact it doesn't preserve the value of the expression for any $x$).
Intuitively, if you subtract $6$ less than the $9x$, then you should end up with a larger number than you subtract all of the $9x$. But your $-9-9x-6$ will yield less than $-9-9x$.
$\endgroup$
