Let's go through each part of the question in detail:
### First Part: Integer Operations
We are given two integers, [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. We need to determine which of the following operations may NOT yield an integer:
(a) [tex]\(a + b\)[/tex]
(b) [tex]\(a - b\)[/tex]
(c) [tex]\(a \div b\)[/tex] (note this presumably refers to the division operation [tex]\(a / b\)[/tex])
(d) [tex]\(a \times b\)[/tex]
1. Addition ([tex]\(a + b\)[/tex]):
- Adding two integers always results in an integer.
- For example, if [tex]\(a = 3\)[/tex] and [tex]\(b = 2\)[/tex], then [tex]\(3 + 2 = 5\)[/tex].
2. Subtraction ([tex]\(a - b\)[/tex]):
- Subtracting one integer from another always results in an integer.
- Using our example, [tex]\(3 - 2 = 1\)[/tex].
3. Division ([tex]\(a / b\)[/tex]):
- Dividing one integer by another does not necessarily result in an integer. This operation may yield a fraction or a decimal.
- For example, [tex]\(3 / 2 = 1.5\)[/tex], which is not an integer.
4. Multiplication ([tex]\(a \times b\)[/tex]):
- Multiplying two integers always results in an integer.
- Using our example, [tex]\(3 \times 2 = 6\)[/tex].
Therefore, the operation that may NOT be an integer is:
(c) [tex]\(a \div b\)[/tex].
### Second Part: Multiplications resulting in specified products
We are asked to determine which of the following expressions is NOT equal to [tex]\( (-11) \times 9 \)[/tex]:
(a) [tex]\(11 \times (-9)\)[/tex]
(b) [tex]\(-(11 \times 9)\)[/tex]
(c) [tex]\((-11) \times (-9)\)[/tex]
(d) [tex]\(9 \times (-11)\)[/tex]
1. Calculate [tex]\( (-11) \times 9 \)[/tex]:
- [tex]\((-11) \times 9 = -99\)[/tex]
2. Evaluate each option:
(a) [tex]\(11 \times (-9)\)[/tex]:
- [tex]\(11 \times (-9) = -99\)[/tex]
- This is equal to [tex]\((-11) \times 9\)[/tex].
(b) [tex]\(-(11 \times 9)\)[/tex]:
- [tex]\(11 \times 9 = 99\)[/tex]
- [tex]\(-(99) = -99\)[/tex]
- This is equal to [tex]\((-11) \times 9\)[/tex].
(c) [tex]\((-11) \times (-9)\)[/tex]:
- [tex]\((-11) \times (-9) = 99\)[/tex]
- This is NOT equal to [tex]\((-11) \times 9\)[/tex].
(d) [tex]\(9 \times (-11)\)[/tex]:
- [tex]\(9 \times (-11) = -99\)[/tex]
- This is equal to [tex]\((-11) \times 9\)[/tex].
Therefore, the expression that is NOT equal to [tex]\( (-11) \times 9 \)[/tex] is:
(c) [tex]\((-11) \times (-9)\)[/tex].
### Summary:
1. The operation which may NOT be an integer: (c) [tex]\(a \div b\)[/tex].
2. [tex]\( (-11) \times 9 \)[/tex] is NOT equal to: (c) [tex]\((-11) \times (-9)\)[/tex].
So the answers are:
1. [tex]\(3\)[/tex]
2. [tex]\(3\)[/tex]
