Question

1. represent real - world problems a poll of likely voters shows that the incumbent will get 51% of the vote in an upcoming election. based on the number of voters polled, the results of the poll could be off by as much as 3 percentage points. what does this mean for the incumbent?
2. explain the error a student solved the inequality $|x - 1|-3>1$ graphically. identify and correct the students error.

i graphed the functions $f(x)=|x - 1|-3$ and $g(x)=1$. because the graph of $g(x)$ lies above the graph of $f(x)$ between $x = - 3$ and $x = 5$, the solution of the inequality is $-3

Explanation:

For Question 22:

Step1: Define margin of error

The poll has a ±3% margin.

Step2: Calculate minimum vote share

Subtract margin from 51%:
$51\% - 3\% = 48\%$

Step3: Calculate maximum vote share

Add margin to 51%:
$51\% + 3\% = 54\%$

Step1: Identify student's error

The inequality is $|x-1|-3 > 1$, which means we need where $f(x) = |x-1|-3$ is above $g(x)=1$, not below. The student reversed the relationship.

Step2: Rewrite the inequality

Isolate the absolute value:
$|x-1| > 1 + 3$
$|x-1| > 4$

Step3: Solve absolute value inequality

Split into two cases:
Case1: $x-1 > 4$ → $x > 5$
Case2: $x-1 < -4$ → $x < -3$

Answer:

The incumbent's actual vote share is expected to be between 48% and 54% of the vote.

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For Question 23: