Question

to win a contest, the number of beans in a jar has to be guessed within 20 of the actual number. if the number of beans in the jar is 645, which equation can be used to find the minimum and maximum number of beans that will win the contest, and what is the maximum guess that could win?

$|x - 645| = 20$; maximum guess: 665 beans

$|x - 645| = 20$; maximum guess: 635 beans

$|x - 20| = 645$; maximum guess: 665 beans

$|x - 20| = 645$; maximum guess: 635 beans

Explanation:

Step1: Understand absolute value equation

The absolute value equation \(|x - 645| = 20\) represents the situation where the difference between the guess \(x\) and the actual number \(645\) is at most \(20\). This means \(x-645 = 20\) or \(x - 645=- 20\).

Step2: Solve for maximum guess

To find the maximum guess, we solve \(x-645 = 20\).
Adding \(645\) to both sides of the equation: \(x=645 + 20=665\).
We can check the other options: For \(|x - 645| = 20\), solving \(x - 645=-20\) gives \(x = 625\) (minimum), and \(x = 665\) (maximum). The second option has a wrong maximum (635 is wrong as \(645-20 = 625\) and \(645 + 20=665\)). The equations \(|x - 20|=645\) are incorrect as they represent the difference between guess and 20 is 645, which is not the case here. The fourth option also has wrong equation and wrong maximum.

Answer:

The correct option is the first one: \(|x - 645| = 20\); maximum guess: 665 beans