Problem 112 - Elementary and Intermediate Algebra

Question

Think of two numbers. First, add the absolute value of the two numbers, and write your answer. Second, add the two numbers, take the absolute value of that sum, and write that answer. Do the two answers agree? Can you find two numbers that produce different answers? When do you get answers that agree, and when don't you?

Step-by-Step Solution

Answer

Results agree when both numbers have the same sign or one is zero. Otherwise, they differ.

Step 1: Define Two Numbers

Let's select two numbers. For example, let \( a = -3 \) and \( b = 5 \).

Step 2: Calculate the Absolute Value of Each Number and Sum Them

Calculate the absolute values: \( |a| = |-3| = 3 \) and \( |b| = |5| = 5 \).Now, sum these absolute values: \( 3 + 5 = 8 \).

Step 3: Sum the Two Numbers and Take the Absolute Value

Sum the numbers first: \( a + b = -3 + 5 = 2 \).Now take the absolute value of the sum: \( |2| = 2 \).

Step 4: Compare the Two Results

Compare the two results: From Step 2, we have 8, and from Step 3, we have 2. These results do not agree.

Step 5: Determine When Results Agree

For results to agree, the absolute value operation must not be necessary after summing. This occurs if both numbers have the same sign (positive or negative), or if one or both are zero.

Step 6: Conclusion on Different Results

Differences occur when two numbers have opposite signs and their absolute values differ significantly. In this situation, \(|a| + |b| e |a + b|\).

Key Concepts

Addition of Integers

Adding integers can sometimes be confusing, especially when dealing with both positive and negative numbers. Here's a simple way to approach it:
First, understand the rule:

• If the numbers have the same sign (both positive or both negative), add their absolute values and assign the common sign to the result.
• If the numbers have different signs, subtract the smaller absolute value from the larger one and give the result the sign of the number with the larger absolute value.

Absolute value is crucial here, because it allows us to focus on the magnitude of numbers without worrying about their signs. For example, to add a) -3 and 5:

• The absolute value of -3 is 3.
• The absolute value of 5 is 5.

The numbers have different signs, so we subtract: 5 - 3 = 2, and since 5 is larger and positive, the result is +2.

Comparing Sums

Comparing sums means looking at two different results of addition to see if they match. In this exercise, we compared

• The sum of the absolute values: |a| + |b|
• The absolute value of the sum: |a + b|

These can give different answers when numbers have opposite signs. The contrast occurs because when numbers are negative, their summation may reduce or cancel their individual magnitudes.
For instance:

• Summing |(-3)| + |5| gives 8
• However, summing (-3) + 5 gives a net value of 2

Here, the results differ because canceling effects of opposite signs leads to a smaller absolute resultant sum. Identifying this scenario can be crucial in understanding how changing signs affects sums.

Algebraic Reasoning

Algebraic reasoning helps us understand patterns and results when working with numbers and their operations. It's a method of problem-solving and justifying why certain things hold true.
When considering absolute values and sums, think about the properties of numbers:

• Why does |a| + |b| not always equal |a + b|?
• Answers align when numbers are of the same sign.
• The discrepancy illustrates how positive values are additive in nature, while negatives can cancel out.

To fully "reason out" this problem, observe:

• If either number is zero, adding does not change its absolute nature.
• If both numbers share a sign, their sum naturally reflects their individual addition before modulus.
• Opposite signs create a tug-of-war in magnitude, decreasing absolute sums when recalculated post-addition.

Use algebraic reasoning to predict outcomes based on signs and their interactions in arithmetic processes.