Let's discuss each pair of expressions and determine if they are equal in value.
1. First Pair:
[tex]\[
\lfloor-6\rfloor \quad \text{and} \quad \lceil-6\rceil
\][/tex]
- The floor function, [tex]\(\lfloor x \rfloor\)[/tex], gives the greatest integer less than or equal to [tex]\(x\)[/tex].
- The ceiling function, [tex]\(\lceil x \rceil\)[/tex], gives the smallest integer greater than or equal to [tex]\(x\)[/tex].
For [tex]\(x = -6\)[/tex]:
- [tex]\(\lfloor -6 \rfloor\)[/tex] is the greatest integer less than or equal to [tex]\(-6\)[/tex], which is [tex]\(-6\)[/tex].
- [tex]\(\lceil -6 \rceil\)[/tex] is the smallest integer greater than or equal to [tex]\(-6\)[/tex], which also is [tex]\(-6\)[/tex].
Therefore, [tex]\(\lfloor -6 \rfloor = -6\)[/tex] and [tex]\(\lceil -6 \rceil = -6\)[/tex]. Thus, the first pair of expressions are equal in value:
[tex]\[
\lfloor-6\rfloor \quad \text{and} \quad \lceil-6\rceil \quad \text{are equal}.
\][/tex]
2. Second Pair:
[tex]\[
\lceil-3.2\rceil \quad \text{and} \quad \lceil-2.6\rceil
\][/tex]
- The ceiling function, [tex]\(\lceil x \rceil\)[/tex], gives the smallest integer greater than or equal to [tex]\(x\)[/tex].
For [tex]\(x = -3.2\)[/tex] and [tex]\(x = -2.6\)[/tex]:
- [tex]\(\lceil -3.2 \rceil\)[/tex] is the smallest integer greater than or equal to [tex]\(-3.2\)[/tex], which is [tex]\(-3\)[/tex].
- [tex]\(\lceil -2.6 \rceil\)[/tex] is the smallest integer greater than or equal to [tex]\(-2.6\)[/tex], which is [tex]\(-2\)[/tex].
Therefore, [tex]\(\lceil -3.2 \rceil = -3\)[/tex] and [tex]\(\lceil -2.6 \rceil = -2\)[/tex]. Thus, the second pair of expressions are not equal in value:
[tex]\[
\lceil-3.2\rceil \quad \text{and} \quad \lceil-2.6\rceil \quad \text{are not equal}.
\][/tex]
In conclusion:
- [tex]\(\lfloor-6\rfloor\)[/tex] and [tex]\(\lceil-6\rceil\)[/tex] are equal.
- [tex]\(\lceil-3.2\rceil\)[/tex] and [tex]\(\lceil-2.6\rceil\)[/tex] are not equal.
