Question

1. let s = {x, 1, 7} where the mean of s is also the median of s. find the product of all possible x

Explanation:

Step1: Analyze cases for median

We have set \( S=\{X, 1, 7\} \). The median of a set with three elements depends on the value of \( X \). We consider three cases:

Case 1: \( X\leq1 \)

If \( X\leq1 \), then the ordered set is \( \{X, 1, 7\} \). The median is \( 1 \). The mean of the set is \( \frac{X + 1+7}{3}=\frac{X + 8}{3} \). Since mean = median, we have:
\( \frac{X + 8}{3}=1 \)
Multiply both sides by 3: \( X+8 = 3 \)
Subtract 8 from both sides: \( X=3 - 8=- 5 \)

Case 2: \( 1

If \( 1 < X<7 \), then the ordered set is \( \{1, X, 7\} \). The median is \( X \). The mean is \( \frac{X + 1+7}{3}=\frac{X + 8}{3} \). Setting mean = median:
\( \frac{X + 8}{3}=X \)
Multiply both sides by 3: \( X + 8=3X \)
Subtract \( X \) from both sides: \( 8 = 2X \)
Divide both sides by 2: \( X = 4 \)

Case 3: \( X\geq7 \)

If \( X\geq7 \), then the ordered set is \( \{1, 7, X\} \). The median is \( 7 \). The mean is \( \frac{X + 1+7}{3}=\frac{X + 8}{3} \). Setting mean = median:
\( \frac{X + 8}{3}=7 \)
Multiply both sides by 3: \( X + 8=21 \)
Subtract 8 from both sides: \( X=21 - 8 = 13 \)

Step2: Find the product of all possible \( X \)

The possible values of \( X \) are \( - 5,4,13 \). The product of these values is \( (-5)\times4\times13=-260 \)

Answer:

\(-260\)