42. Hosni gave the following argument that 2(a 1 b) 5
2a 1 2b, for all integers a and b. If the argument is
correct, supply the missing reasons. If it is incorrect,
explain why not.
-a (b+a)= (-1)(a+b)
= (-1)a+ (-1) b
= a+ -b
To determine if the argument is correct, let's examine the steps and check if the reasoning is accurate.
1. The argument starts with the expression -a (b + a).
2. It then claims that -a (b + a) is equal to (-1)(a + b).
The missing reason here appears to be the distributive property of multiplication, which states that for any integers a, b, and c:
a(b + c) = ab + ac
Using this property, we can apply it to the expression -a (b + a):
-a (b + a) = (-1)(b + a)a = (-1)(a + b)a
Now, let's simplify the next step of the argument:
(-1)(a + b)a = (-1)(a) + (-1)(b) = -a - b
However, the argument mistakenly claims that -a (b + a) is equal to a + -b, which is inconsistent with the previous step.
Therefore, the argument is incorrect. The mistake lies in the final simplification step, where the signs of the terms are not properly accounted for. The correct final result should be -a - b, not a + -b.