Problem 82 - Elementary Algebra

Question

Find the distance between the given numbers on a number line. $$ 10.7 \text { and }-2.8 $$

Step-by-Step Solution

Answer

The distance is 13.5 units.

Step 1: Understand the Problem

We need to find the distance between two points on a number line. These points are 10.7 and -2.8.

Step 2: Apply the Distance Formula

Remember that the distance between two points on a number line, say A and B, is given by the absolute value of their difference: \[ d = |A - B| \] In our case, A is 10.7 and B is -2.8.

Step 3: Substitute the Values

Substitute the values of A and B into the formula: \[ d = |10.7 - (-2.8)| \] This simplifies to: \[ d = |10.7 + 2.8| \]

Step 4: Calculate the Result

Calculate the value inside the absolute value: \[ 10.7 + 2.8 = 13.5 \] Then take the absolute value, which is simply 13.5 in this case since it's positive.

Key Concepts

Distance on a Number Line

When we talk about distance on a number line, we're referring to the length between two points that are positioned along the number line. Basically, it's like asking how many steps it takes to get from one point to another. Distance on a number line is always non-negative because length can't be negative. To find this distance, we use a simple method: subtract one value from the other, and then take the absolute value of the result.

• Identify the two points you're working with.
• Calculate their difference by subtracting one from the other.
• Apply the absolute value to get the actual distance.

This ensures the distance is always a positive number, showing its true length, irrespective of which point comes first in the subtraction.

Absolute Value

The concept of absolute value is crucial in mathematics, especially when dealing with real numbers. It refers to the distance of a number from zero on a number line, disregarding any negative sign. Absolute value is always a non-negative number because distance can't be negative.
Absolute value is represented by vertical bars, for example, the absolute value of -3 is written as \(|-3|\), and it's equal to 3.

• If the number is positive or zero, its absolute value is the number itself.
• If the number is negative, its absolute value is the opposite of the number.

This concept helps in finding the distance between numbers, as it ensures the result represents a real span on the number line.

Real Numbers

Real numbers encompass all the numbers that can be found on a number line, covering everything from the negatives through zero to the positives. They include various subcategories such as whole numbers, integers, rational numbers, and irrational numbers.

• Whole numbers include zero and all positive numbers without fractions.
• Integers extend whole numbers to include negative counterparts.
• Rational numbers can be expressed as fractions, where both numerator and denominator are integers.
• Irrational numbers can't be neatly expressed as fractions; examples include 𝜋 and the square root of 2.

Understanding real numbers and their structure aids problem-solving by allowing us to manipulate their values using established mathematical properties.

Problem Solving Steps

Solving mathematical problems often involves a series of logical steps to achieve the correct result. This structured approach can be applied to various types of problems, including those dealing with distances between numbers on a number line.
Here's a typical set of problem-solving steps:

• Understand the problem: Grasp what is being asked. This often involves reading the problem carefully and identifying known variables.
• Apply relevant formulas: For instance, the distance formula when dealing with points on a number line.
• Substitute the given values: Insert known values into the formula.
• Calculate and simplify: Do the necessary arithmetic to find the solution, ensuring you check your work for errors.

By following these steps, you can tackle problems in an organized and efficient manner, minimizing mistakes and ensuring a thorough understanding.