Question

Set \(S\) contains the elements \(\\{-4, x, 0,17,1\\} .\) What is the median of set \(S ?\) (1) \(x>1\) (2) \(x<2\)

Short answer

Median is 1 for condition (1) and 0 for condition (2).

Step-by-step solution

- Understand the Problem

The problem asks for the median of set \(S\) which contains \{-4, x, 0, 17, 1\}. We need to determine the median based on the additional constraints given in statements (1) and (2).

- Arrange the Set in Ascending Order

To find the median, first arrange the elements of set \(S\) in ascending order. Since there is an unknown element \(x\), consider how it could be positioned based on its value.

- Analysis of Condition (1)

Given that \(x > 1\), the set becomes \{-4, 0, 1, 17, x\}. Since \(x > 1\), \(x\) will be greater than 1, 0, and -4 but less than 17. Thus, the ordered set is \{-4, 0, 1, x, 17\}. For an ordered set of 5 numbers, the median is the third number. Therefore, the median is 1 when \(x > 1\).

- Analysis of Condition (2)

Given that \(x < 2\), the set can be \{-4, x, 0, 1, 17\}. Depending on the value of \(x\) (which can range from a minimal negative number up to just below 2), \(x\) could be the smallest (if negative), fit between -4 and 0, or between 0 and 1. The median is still the third element in the ordered set. Therefore, the median is 0 if -4 < x < 0, and if 0 ≤ x < 1, the median is 0.

Key concepts

Understanding Median Calculation

Calculating the median might sound intimidating, but it’s simpler than you think! The median is the middle value when all elements of a set are arranged in ascending order. It divides the set into two equal halves. For instance, in a set of five numbers, the median is the third number. When the number of elements is odd, the median is the middle element. If the number is even, the median is the average of the two middle values. To find the median accurately, follow these steps:

• First, arrange the numbers in ascending order.
• Next, locate the middle element.
• If there are an even number of elements, calculate the average of the two middle elements.

If we apply these principles to the given problem, finding the median becomes straightforward.

Arranging Elements in Ascending Order

To find the median, sorting the elements in ascending order is crucial. Ascending order means arranging the numbers from smallest to largest. This rearrangement helps us identify the middle value accurately. In the problem, the set is \{-4, x, 0, 17, 1\}. Depending on the value of 'x', the position of each element might change. Here's how you can visualize it clearly:

• If \( x > 1 \), then 'x' is greater than 1 but less than 17, making the sorted set \{-4, 0, 1, x, 17\}.
• For \( x < 2 \), 'x' can be anywhere between a highly negative number to just below 2. So, the position could be after -4 and before 17, resulting in sets like \{-4, x, 0, 1, 17\}.

Once the elements are in order, identifying the middle one becomes easy. This process highlights the importance of sorting and how it leads us to the correct median.

Introduction to Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. Understanding the basics of set theory can simplify many complex problems like the one we are solving. A 'set' is simply a collection of distinct objects. For example, set \( S \) in our problem is \{-4, x, 0, 17, 1\}. Here’s how set theory can help:

• Elements: The components of the set are called 'elements'. In our set, the elements are -4, 0, 17, 1, and 'x'.
• Notation: Sets are usually denoted with curly braces \{ \}. For instance, set \( S \) is written as \{ -4, x, 0, 17, 1 \}.
• Order: The order in a set does not matter unless we are sorting it to find the median.

Understanding these fundamentals aids in structuring our approach effectively. For instance, by recognizing that any element ('x') has a specific role within the set, we determine how changing its value affects the median. Combining knowledge of sorting and set theory offers a powerful toolkit for solving similar GMAT questions.