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Chapter 0: Problem 39

Find the midpoint of the given interval. $$ \left[-\frac{1}{2}, \frac{3}{4}\right] $$

Short Answer

Expert verified
The midpoint of the given interval is \(\frac{1}{8}\).

Step by step solution

01

Identify the Endpoints

Identify the given endpoints of the interval. In this problem, the interval is \(\left[-\frac{1}{2}, \frac{3}{4}\right]\), So, the endpoints are \(End_1 = -\frac{1}{2}\) and \(End_2 = \frac{3}{4}\).
02

Plug into the Midpoint Formula

Plug the identified endpoints into the midpoint formula. So, \(Midpoint = \frac{End_1 + End_2}{2} = \frac{-\frac{1}{2} + \frac{3}{4}}{2}\).
03

Perform the Calculation

Perform the calculation. So, the required midpoint is \(\frac{-1/2 + 3/4}{2} = \frac{1/4}{2} = \frac{1}{8}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Endpoints
Endpoints are crucial for understanding intervals. They are the numbers that mark the start and end of an interval on the number line. In this exercise, we have two endpoints:
  • The first endpoint is \( End_1 = -\frac{1}{2} \).
  • The second endpoint is \( End_2 = \frac{3}{4} \).
These values describe the specific section of the number line we are working with. Identifying the endpoints of an interval is the foundational step before applying any formula, like the midpoint formula. Knowing the endpoints allows you to calculate everything else needed for solving an interval question. Take your time to ensure you have the correct numbers.
Midpoint Formula
The midpoint formula helps us find the center point of an interval between two endpoints. It鈥檚 like finding a balance point between two numbers. The formula is given by:\[Midpoint = \frac{End_1 + End_2}{2}\]For the interval \( \left[-\frac{1}{2}, \frac{3}{4}\right] \), we plug in the values:
  • Use \( End_1 = -\frac{1}{2} \)
  • Use \( End_2 = \frac{3}{4} \)
  • Calculate: \( \frac{-\frac{1}{2} + \frac{3}{4}}{2} \)
  • Simplify: This becomes \( \frac{1/4}{2} = \frac{1}{8} \)
So, the midpoint is \( \frac{1}{8} \). This formula is very useful in various mathematical and real-world applications where you need to find a central point.
Interval Notation
Interval notation is a way of writing subsets of the real number line. It gives you a quick reference for which numbers are included in an interval. There are two types of brackets:
  • Square brackets \([ ]\) include the endpoint values.
  • Round brackets \(( )\) exclude the endpoint values.
For our problem, the interval \( \left[-\frac{1}{2}, \frac{3}{4}\right] \) uses square brackets. This means both \(-\frac{1}{2}\) and \(\frac{3}{4}\) are included in the interval. Understanding interval notation can be very helpful for solving equations and inequalities, as it precisely indicates which numbers are included. Always pay attention to the type of bracket used in interval notation.

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