Problem 138 - Prealgebra and Introductory Algebra

Question

Point \(A\) is a point on the number line halfway between \(-9\) and \(3 .\) Point \(B\) is a point halfway between \(A\) and the graph of 1 on the number line. Point \(B\) is the graph of what number?

Step-by-Step Solution

Answer

Point B is the graph of the number -1.

Step 1: Determine the Location of Point A

To find the number that is halfway between -9 and 3, compute the average of -9 and 3. To calculate the average, add the two numbers together and then divide by 2. \[A = \frac{-9 + 3}{2} = -3 \] So, point \(A\) is at -3.

Step 2: Determine the Location of Point B

Now, find the number which is halfway between -3 (the location of Point A) and 1. Use the same approach as in Step 1 - add the two numbers together and then divide by 2. \[B = \frac{-3 + 1}{2} = -1 \] So, point \(B\) is at -1.

Key Concepts

Finding the Midpoint

When we talk about the midpoint on a number line, we're referring to the point that is exactly in the center between two values. Think of it as the halfway mark in a race: it's the point where you've covered an equal distance from the start to the end.

Finding this midpoint is a valuable prealgebra skill, especially in geometry and coordinate graphing. To locate the midpoint between two numbers, you calculate the average of these numbers. This involves adding the two numbers together and dividing the result by 2. For example, if you wanted to find the midpoint between -9 and 3, you would compute \(\frac{-9 + 3}{2}\) to get -3. This means that -3 is equidistant from both -9 and 3 on the number line. Understanding this concept is crucial for visualizing and solving more complex algebraic problems later on.

Average of Numbers

Averaging numbers is a fundamental concept in mathematics that is often introduced in early education, and it's used extensively in various fields and daily life. To find the average of a set of numbers, you add all the numbers together, then divide by the number of items you added. This gives you the mean, or the typical value in your set.

For example, if you received scores of 90, 80, and 70 on three tests, the average score would be calculated as \(\frac{90 + 80 + 70}{3}\) which equals 80. In the context of number lines, averaging two numbers gives you their midpoint. So when the number line exercise asks you to find a point halfway between two others, you're essentially being asked to calculate the average of those two points. The average helps you understand and interpret sets of data by providing a single representative value.

Prealgebra Concepts

Prealgebra serves as the foundation of your journey into the world of algebras, such as solving equations, understanding functions, and analyzing geometric shapes. This preliminary mathematical stage includes understanding number operations, the significance of the number line, and the ability to calculate averages and midpoints, which are stepping stones to algebraic thinking.

In dealing with prealgebra, crucial skills include being comfortable with negative numbers and understanding how to find points on the number line. These skills are put into practice in our exercise, where we find points A and B by averaging. Grasping these concepts ensures a smoother transition into algebra, where the same ideas are expanded upon with variables and more complex operations. Therefore, fostering a strong grasp of these prealgebra concepts will support future success in mathematics and the application of these principles in real-world situations.