Problem 35 - Algebra for College Students

Question

Perform the following operations with real numbers. $$-21.4-(-14.9)$$

Step-by-Step Solution

Answer

The result of \(-21.4 - (-14.9)\) is \(-6.5\).

Step 1: Identify the Operation

Look at the expression \(-21.4 - (-14.9)\). It involves subtraction of a negative number. When you subtract a negative number, it is equivalent to adding the positive of that number.

Step 2: Convert Subtraction of Negative

Rewrite the subtraction \(-21.4 - (-14.9)\) as an addition:\(-21.4 + 14.9\).

Step 3: Perform the Addition

Add the numbers \(-21.4 + 14.9\). Start by looking at the numbers: 1. Calculate \(21.4 - 14.9\), which gives \(6.5\).2. Since you are adding a negative and a positive number, the result will be negative: \(-6.5\).

Key Concepts

Subtraction of Negative Numbers

The subtraction of negative numbers often confuses students, but it can be understood simply, with a bit of logic and practice. When you subtract a negative number, you are essentially adding a positive version of that number. This is because negatives and negatives cancel each other out, similar to how subtracting two negatives results in a positive. For example, in the original exercise, we have:

• Starting with \(-21.4 - (-14.9)\).
• We notice that subtracting \(-14.9\) is the same as adding \(+14.9\).
• So, this becomes \(-21.4 + 14.9\).

Remember, whenever you see subtraction of a negative number, just change it to addition. This keeps arithmetic straightforward and makes solving such problems easier.

Adding Positive and Negative Numbers

Adding positive and negative numbers can be tricky at first, but understanding the basic rules can make it simpler. In mathematics, when you add a positive and a negative number, you are essentially calculating the difference between the two values, taking their signs into account.Here's a breakdown to clarify:

• Look at the step from the exercise: \(-21.4 + 14.9\).
• You can think of this as combining a negative quantity with a positive quantity.
• First, find the absolute values: \(21.4\) and \(14.9\).
• Calculate the difference: \(21.4 - 14.9 = 6.5\).
• Since the larger absolute value was \(21.4\) (which originally had the negative sign), the result is negative: \(-6.5\).

The key here is to subtract the smaller absolute value from the larger and give the result the sign of the number with the larger absolute value.

Properties of Real Numbers

Understanding the properties of real numbers helps in simplifying arithmetic operations and solving equations effectively. Real numbers include all the numbers on the number line, covering both positive and negative integers, fractions, and decimals.Here are some useful properties:

• **Commutative Property (of Addition):** \(a + b = b + a\). It simply means the order of addition does not matter.
• **Associative Property (of Addition):** \((a + b) + c = a + (b + c)\). You can group numbers differently when adding.
• **Additive Identity Property:** The sum of any number and 0 is the number itself, \(a + 0 = a\).
• **Additive Inverse Property:** The sum of a number and its negative (inverse) is \(0\), \(a + (-a) = 0\).

These properties provide the framework for working with different operations, making calculations easier to handle. They ensure consistent outcomes when performing arithmetic with real numbers, regardless of the complexity of the problem.