Answer :
Sure, let’s break this down step-by-step, answering each question thoroughly:
1. Reason abstractly about odd integers and even integers:
We need to prove that any odd integer can be expressed as the sum of an even integer and an odd integer.
- An even integer can be written in the form [tex]\(2n\)[/tex] for some integer [tex]\(n\)[/tex].
- An odd integer can be written in the form [tex]\(2m + 1\)[/tex] for some integer [tex]\(m\)[/tex].
Consider an odd integer [tex]\(k\)[/tex]. By definition, we can express [tex]\(k\)[/tex] as [tex]\(k = 2m + 1\)[/tex] (for some integer [tex]\(m\)[/tex]).
Let's write [tex]\(k\)[/tex] as a sum of two numbers:
- Let [tex]\(2n\)[/tex] be an even number.
- Let [tex]\(2r + 1\)[/tex] be an odd number.
Choose [tex]\(2n\)[/tex] (even number) to be any even number less than [tex]\(k\)[/tex], and the difference will be:
[tex]\[ 2m + 1 = 2n + (2(m-n) + 1) \][/tex]
Here, [tex]\(2n\)[/tex] is an even integer, and [tex]\(2(m-n) + 1\)[/tex] is an odd integer. Thus, any odd integer can indeed be expressed as the sum of an even and an odd integer.
2. Solve the equation [tex]\(x - 5 = -2\)[/tex]:
Follow these steps to solve for [tex]\(x\)[/tex] systematically:
1. Given the equation [tex]\(x - 5 = -2\)[/tex].
2. To isolate [tex]\(x\)[/tex], add [tex]\(5\)[/tex] to both sides of the equation:
[tex]\[ x - 5 + 5 = -2 + 5 \][/tex]
3. Simplify both sides:
[tex]\[ x = 3 \][/tex]
Thus, the solution to the equation [tex]\(x - 5 = -2\)[/tex] is [tex]\(x = 3\)[/tex].
3. Determine if Andre's statement is a conjecture or a theorem based on the pattern provided:
The statement is: "The number of sides of a polygon is equal to its number of angles."
- From the table provided:
[tex]\[ \begin{tabular}{|l|c|c|} \hline Polygon & Sides & Angles \\ \hline Triangle & 3 & 3 \\ \hline Quadrilateral & 4 & 4 \\ \hline Pentagon & 5 & 5 \\ \hline \end{tabular} \][/tex]
We observe that for each polygon, the number of sides is equal to the number of angles.
In geometry, it is a known fact that any simple polygon (including triangles, quadrilaterals, pentagons, etc.) has an equal number of sides and angles. Hence, Andre's observation confirms a well-established geometric theorem rather than a conjecture. A theorem is a statement that has been proven on the basis of previously established statements and logical reasoning. Therefore, Andre’s statement is a theorem.
4. Evaluate the conclusion drawn by an investigator at a crime scene:
The problem mentions that hairs found at a crime scene are consistent with those of a suspect. Based on this evidence, the investigator concludes that the suspect was at the crime scene.
Let’s examine this:
- The evidence aligns with the characteristics of the suspect.
- However, 'consistent with' only indicates a match in certain characteristics, it does not confirm the presence beyond doubt.
To definitively conclude that the suspect was at the crime scene solely based on the hair evidence would be premature without further supporting evidence. The investigator's conclusion is a hypothesis that requires more thorough investigation and additional evidence to ensure it’s not a coincidence and to rule out other potential explanations (e.g., secondary transfer, presence of other matching individuals, etc.).
This covers a detailed explanation for each problem presented.
1. Reason abstractly about odd integers and even integers:
We need to prove that any odd integer can be expressed as the sum of an even integer and an odd integer.
- An even integer can be written in the form [tex]\(2n\)[/tex] for some integer [tex]\(n\)[/tex].
- An odd integer can be written in the form [tex]\(2m + 1\)[/tex] for some integer [tex]\(m\)[/tex].
Consider an odd integer [tex]\(k\)[/tex]. By definition, we can express [tex]\(k\)[/tex] as [tex]\(k = 2m + 1\)[/tex] (for some integer [tex]\(m\)[/tex]).
Let's write [tex]\(k\)[/tex] as a sum of two numbers:
- Let [tex]\(2n\)[/tex] be an even number.
- Let [tex]\(2r + 1\)[/tex] be an odd number.
Choose [tex]\(2n\)[/tex] (even number) to be any even number less than [tex]\(k\)[/tex], and the difference will be:
[tex]\[ 2m + 1 = 2n + (2(m-n) + 1) \][/tex]
Here, [tex]\(2n\)[/tex] is an even integer, and [tex]\(2(m-n) + 1\)[/tex] is an odd integer. Thus, any odd integer can indeed be expressed as the sum of an even and an odd integer.
2. Solve the equation [tex]\(x - 5 = -2\)[/tex]:
Follow these steps to solve for [tex]\(x\)[/tex] systematically:
1. Given the equation [tex]\(x - 5 = -2\)[/tex].
2. To isolate [tex]\(x\)[/tex], add [tex]\(5\)[/tex] to both sides of the equation:
[tex]\[ x - 5 + 5 = -2 + 5 \][/tex]
3. Simplify both sides:
[tex]\[ x = 3 \][/tex]
Thus, the solution to the equation [tex]\(x - 5 = -2\)[/tex] is [tex]\(x = 3\)[/tex].
3. Determine if Andre's statement is a conjecture or a theorem based on the pattern provided:
The statement is: "The number of sides of a polygon is equal to its number of angles."
- From the table provided:
[tex]\[ \begin{tabular}{|l|c|c|} \hline Polygon & Sides & Angles \\ \hline Triangle & 3 & 3 \\ \hline Quadrilateral & 4 & 4 \\ \hline Pentagon & 5 & 5 \\ \hline \end{tabular} \][/tex]
We observe that for each polygon, the number of sides is equal to the number of angles.
In geometry, it is a known fact that any simple polygon (including triangles, quadrilaterals, pentagons, etc.) has an equal number of sides and angles. Hence, Andre's observation confirms a well-established geometric theorem rather than a conjecture. A theorem is a statement that has been proven on the basis of previously established statements and logical reasoning. Therefore, Andre’s statement is a theorem.
4. Evaluate the conclusion drawn by an investigator at a crime scene:
The problem mentions that hairs found at a crime scene are consistent with those of a suspect. Based on this evidence, the investigator concludes that the suspect was at the crime scene.
Let’s examine this:
- The evidence aligns with the characteristics of the suspect.
- However, 'consistent with' only indicates a match in certain characteristics, it does not confirm the presence beyond doubt.
To definitively conclude that the suspect was at the crime scene solely based on the hair evidence would be premature without further supporting evidence. The investigator's conclusion is a hypothesis that requires more thorough investigation and additional evidence to ensure it’s not a coincidence and to rule out other potential explanations (e.g., secondary transfer, presence of other matching individuals, etc.).
This covers a detailed explanation for each problem presented.
