Hosni gave the following argument that -(a+b)=-a+-b for all integers a and b. If the argument is correct, supply the missing reasons. If it is incorrect, explain why. -(a+b)=(-1)(a+b) =(-1)a+(-1)b =-a+-b -(a+b)=(-1)(a+b)=(-1)a+(-1)b=(-a)+(-b)=-a+-b
The given argument states that -(a+b) = -a + -b for all integers a and b. Let's analyze whether this argument is correct:
Firstly, the argument utilizes the distributive property by multiplying -(a+b) by -1. This step is correct.
-(a+b) = (-1)(a+b)
Next, the argument applies the distributive property again to expand (-1)(a+b) to (-1)a + (-1)b. This step is also correct.
(-1)(a+b) = (-1)a + (-1)b
Then, the argument rearranges (-1)a and (-1)b in terms of -a and -b, respectively. This step is correct as well.
(-1)a + (-1)b = -a + -b
Therefore, according to this argument, -(a+b) is indeed equal to -a + -b.
To summarize, the argument is correct. The reasons provided are the application of the distributive property twice to equate -(a+b) to -a + -b.