To solve this question, let's first understand what is being asked. We need to represent the true percentage of votes received by the candidate, denoted by [tex]\(x\)[/tex], within a given margin of error.
1. Given Information:
- The estimated percentage of votes: [tex]\(52\%\)[/tex]
- Margin of error: [tex]\(5\%\)[/tex]
2. Understanding the margin of error:
- The margin of error means that the actual percentage of votes [tex]\(x\)[/tex] can be within [tex]\(5\%\)[/tex] above or below the estimated percentage of [tex]\(52\%\)[/tex].
- Mathematically, this can be represented as:
[tex]\[
52\% - 5\% \leq x \leq 52\% + 5\%
\][/tex]
- Simplifying the values:
[tex]\[
47\% \leq x \leq 57\%
\][/tex]
3. Expressing as an absolute value inequality:
- The expression [tex]\(47\% \leq x \leq 57\%\)[/tex] can be written in terms of an absolute value inequality.
- The absolute value inequality allows us to describe the deviation of [tex]\(x\)[/tex] from the estimated percentage [tex]\(52\%\)[/tex].
4. Formulating the absolute value inequality:
- The deviation [tex]\(x\)[/tex] from the estimated [tex]\(52\%\)[/tex] should be no more than [tex]\(5\%\)[/tex]:
[tex]\[
|x - 52\%| \leq 5\%
\][/tex]
- This inequality states that the distance between [tex]\(x\)[/tex] and [tex]\(52\%\)[/tex] should be within [tex]\(5\%\)[/tex] in either direction (above or below).
5. Conclusion:
- The absolute value inequality that correctly represents the interval in which to estimate [tex]\(x\)[/tex] is:
[tex]\[
|x - 52\%| \leq 5\%
\][/tex]
Comparing this with the given options:
- a. [tex]\( |x - 52| \geq 0.05 \)[/tex]
- b. [tex]\( |x - 52| \leq 0.05 \)[/tex]
- c. [tex]\( |x - 0.05| \geq 52 \)[/tex]
- d. [tex]\( |x - 0.05| \leq 52 \)[/tex]
The correct choice is:
b. [tex]\( |x - 52| \leq 0.05 \)[/tex]
