Question

In a sports poll, 53% of those surveyed believe their high school football team will win the state championship. The poll shows a margin of error of 0.5 percentage points. Write and solve an absolute value inequality to find the least and the greatest percent of people that think their team will win the state championship.

Answer 1

See Below

Explanation:

Let total number of people be "n"

Since 53% believe their high school, that would be "0.53n"

Since 0.5% fall in error margin, that would be ± 0.005n

Hence,

Greatest number of people would be:

`((0.53n+0.005n)/(n))*100=0.535`

That is 53.5% of people

and

Least nnumber of people would be:

`((0.53n-0.005n)/(n))*100=0.525`

That is 52.5% of people

Thus, the absolute value equation would be:

0.525n ≤ absolute number of people who think their team will win ≤ 0.535n

Where "n" is the total number of spectators

Answer 2

The required inequality is:
`|x-53|\leq 0.5`

The least and the greatest percent of people that think their team will win the state championship is 52.5% and 53.5% respectively.

Explanation:

Consider the provided information.

In a sports poll, 53% of those surveyed believe their high school football team will win the state championship.

The poll shows a margin of error of 0.5 percentage points.

Let x is the actual number of percent of people that think their team will win the state championship.

The absolute difference of x and 53 should be less than or equal to 0.5

Therefore, the required inequality is:

`|x-53|\leq 0.5`

`\mathrm{Apply\:absolute\:rule}:\quad \mathrm{If}\:|u|\:\le \:a,\:a\:>\:0\:\mathrm{then}\:-a\:\le \:u\:\le \:a`

By using the above rule we can solve the inequality as shown:

`-0.5\le \:x-53\le \:0.5`

Add 53 as shown:

`-0.5+53\le \:x\le \:0.5+53`

`52.5\le \:x\le \:53.5`

Hence, the least and the greatest percent of people that think their team will win the state championship is 52.5% and 53.5% respectively.