Problem 17 - Intermediate Algebra

Question

Perform the operations. See Example 1 . $$ -7.1+2.8 $$

Step-by-Step Solution

Answer

-4.3

Step 1: Identify the Numbers

We are given the expression \(-7.1 + 2.8\). Here, the two numbers involved in the operation are -7.1 and +2.8. It is important to pay attention to the signs.

Step 2: Determine the Operation

The operation between the numbers is addition. Since one number is negative and the other is positive, this effectively becomes a subtraction problem. The numbers involved are 7.1 and 2.8.

Step 3: Subtract the Smaller Number from the Larger

We perform the subtraction by ignoring signs: subtract 2.8 from 7.1. \(7.1 - 2.8 = 4.3\).

Step 4: Determine the Sign of the Result

Since the number with the larger absolute value is -7.1, the result will have the same sign as the larger absolute value. Therefore, the result is \(-4.3\).

Key Concepts

Addition and Subtraction in Integer Operations

In mathematical operations, addition and subtraction are foundational skills, particularly important when dealing with integers. Let's dive into how they work.

When adding integers, you'll often deal with both positive and negative numbers. If two numbers have the same sign, simply add their absolute values, and the result keeps the common sign. For example:

• Adding two positives: \( 3 + 5 = 8 \)
• Adding two negatives: \( -2 + (-4) = -6 \)

However, when the numbers have different signs, the process mimics subtraction. Here, you subtract the smaller absolute value from the larger and take the sign of the number with the larger absolute value. For example, in solving the operation \(-7.1 + 2.8\), we focus on the larger absolute value, 7.1, and thus subtract:

• \( 7.1 - 2.8 = 4.3 \)

The sign of \(-7.1\) ensures the final result is \(-4.3\). So, mastering these operations lies in understanding when to switch between addition and subtraction based on the signs involved.

Understanding Absolute Value

The absolute value of a number represents its distance from zero on a number line, regardless of direction. Absolute values help simplify complex problems, especially when dealing with operations involving different signs.

Here's how to work with absolute values:

• If a number is already positive, its absolute value is the number itself. For example, \(|5| = 5\).
• If a number is negative, its absolute value is the positive equivalent. For instance, \(|-7| = 7\).

Imagine solving \(-7.1 + 2.8\). We compute the absolute values, arriving at 7.1 and 2.8. These values assist in determining which number has a greater influence in the operation, guiding us to subtract the smaller absolute value from the larger. The absolute value thus becomes crucial in understanding how to properly add or subtract numbers, setting aside their signs initially to perform the operation before reapplying the correct sign at the end.

Navigating Positive and Negative Numbers

Positive and negative numbers can often trip up students, but understanding their interaction is key to solving various math problems. Remember:

• Positive numbers are greater than zero and located to the right on a number line.
• Negative numbers are less than zero and sit to the left on a number line.

When performing operations such as \(-7.1 + 2.8\), the interaction between a negative and a positive number turns the problem into one of comparison between their absolute values. The sign of the result hinges on the larger absolute value. Should you write \(a + b\) and observe that \(|a| > |b|\), the sign of the result is that of \(a\). In this way, positive and negative numbers influence not just magnitude but also direction on the number line, giving each operation a unique outcome. By mastering their dynamics, you gain confidence in making accurate calculations even under varying mathematical conditions.