Statistics Question Requires C. Rule?

Question

I do NOT need answers, I need you to provide me with a better understanding of the process to get to the answers. If possible can you provide keywords in the question that lead to why you did what you did. Please do not skip steps or use calculator functions. All I will have available to me is a basic calculator and a table with Z Scores, please check for the rest of my questions:

A recent survey was conducted to compare the cost of solar energy to the cost of gas or electric energy. Results
of the survey revealed that the distribution of the amount of the monthly utility bill of a 3-bedroom house
using gas or electric energy had a mean of $126 and a standard deviation of $14. If nothing is known about the
shape of the distribution, what percentage of homes will have a monthly utility bill of less than $98?

The answer is at most 25% please explain as, best you can, the mental thinking and steps toward getting there.

Answer 1

For this you will need Chebyshev's inequality. It states that for any random variable X having finite mean μ and standard deviation σ and any real number k>0:

P(|X - μ| ≥ kσ) ≤ 1/k².

In this case, letting X be the monthly utility bill, we see that it has mean 126 and standard deviation 14. If X was less than 98, then |X-126| ≥ 28 = 2*14, so we see that:

P(X<98) ≤ P(|X-126| ≥ 2*14) ≤ 1/2² = 1/4

So at most 1/4 of the homes will have a monthly utility bill less than $98.