Sure! Let's address each part of the question in detail.
### a) Find the product of these integers by using a number line.
To find the product of +3 and -4 using a number line, we can perform the operation step by step:
1. Start at 0 on the number line.
2. Moving to the right three steps from 0 (representing +3) would give us +3. However, because we are multiplying by -4, we actually move four groups of -3 (three units left for each of the four steps).
So, imagine each step:
- Starting at 0, move -3 units once: -3
- Move -3 units again: -6
- Move -3 units again: -9
- Move -3 units one last time: -12
Thus, the product of +3 and -4 on the number line is -12.
### b) Does the product verify the closure property of multiplication of integers? Give reason.
The closure property of multiplication states that the product of any two integers is always an integer.
Here, the product of [tex]\( +3 \)[/tex] and [tex]\( -4 \)[/tex] is [tex]\( -12 \)[/tex].
Since -12 is an integer, this verifies that the product of [tex]\( +3 \)[/tex] and [tex]\( -4 \)[/tex] satisfies the closure property of multiplication of integers.
### c) Verify the commutative property of multiplication of integers by using the given integers.
The commutative property of multiplication states that the order in which two numbers are multiplied does not affect their product. In other words, [tex]\( a \times b = b \times a \)[/tex].
To verify the commutative property with the given integers [tex]\( +3 \)[/tex] and [tex]\( -4 \)[/tex]:
- Calculate [tex]\( 3 \times (-4) \)[/tex]:
[tex]\[
3 \times (-4) = -12
\][/tex]
- Calculate [tex]\( (-4) \times 3 \)[/tex]:
[tex]\[
(-4) \times 3 = -12
\][/tex]
Since both calculations give the same result, we have:
[tex]\[
3 \times (-4) = (-4) \times 3 = -12
\][/tex]
Thus, this verifies the commutative property for the given integers [tex]\( +3 \)[/tex] and [tex]\( -4 \)[/tex].
In summary:
a) The product of [tex]\( +3 \)[/tex] and [tex]\( -4 \)[/tex] is [tex]\( -12 \)[/tex].
b) The result [tex]\( -12 \)[/tex] is an integer, thus fulfilling the closure property.
c) The commutative property is verified as [tex]\( 3 \times (-4) = (-4) \times 3 = -12 \)[/tex].
