To determine which option correctly applies the commutative property to the expression [tex]\(3 + (-4)\)[/tex], we need to remember that the commutative property of addition states that the order in which two numbers are added does not change the sum. Mathematically, if [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are two numbers, then [tex]\(a + b = b + a\)[/tex].
Let's evaluate each option:
1. Option 1: [tex]\(3 + (-4) = -3 + 4\)[/tex]\
The left-hand side (LHS) is [tex]\(3 + (-4)\)[/tex], which simplifies to [tex]\(-1\)[/tex]. The right-hand side (RHS) is [tex]\(-3 + 4\)[/tex], which also simplifies to [tex]\(1\)[/tex]. Since [tex]\(-1 \neq 1\)[/tex], this is incorrect.
2. Option 2: [tex]\(3 + (-4) = 3 - 4\)[/tex]\
The LHS is [tex]\(3 + (-4)\)[/tex], which simplifies to [tex]\(-1\)[/tex]. The RHS is [tex]\(3 - 4\)[/tex], which also simplifies to [tex]\(-1\)[/tex]. Although the values match, this does not show the commutative property because we haven't rearranged the order of addition but rather performed subtraction. So this is not a correct application of the commutative property.
3. Option 3: [tex]\(3 + (-4) = -4 + 3\)[/tex]\
The LHS is [tex]\(3 + (-4)\)[/tex], which simplifies to [tex]\(-1\)[/tex]. The RHS is [tex]\(-4 + 3\)[/tex], which also simplifies to [tex]\(-1\)[/tex]. The values match, and the order of the numbers in the addition has been swapped, which correctly demonstrates the commutative property. Therefore, this is correct.
4. Option 4: [tex]\(3 + (-4) = 4 + (-3)\)[/tex]\
The LHS is [tex]\(3 + (-4)\)[/tex], which simplifies to [tex]\(-1\)[/tex]. The RHS is [tex]\(4 + (-3)\)[/tex], which simplifies to [tex]\(1\)[/tex]. Since [tex]\(-1 \neq 1\)[/tex], this is incorrect.
Therefore, the correct application of the commutative property to the expression [tex]\(3 + (-4)\)[/tex] is:
[tex]\[3 + (-4) = -4 + 3\][/tex]
The correct option is [tex]\( \boxed{-4 + 3} \)[/tex].
