Problem 42 - Algebra for College Students

Question

Determine which of the ordered pairs \((1,3),(-2,5)\) \((-6,-4),\) and \((7,-8)\) satisfy each compound or absolute value inequality. $$|x-y|>2$$

Step-by-Step Solution

Answer

The pairs (-2, 5) and (7, -8) satisfy the inequality \(|x-y| > 2\).

Step 1 - Understand the Inequality

The inequality we need to satisfy is \(|x-y|>2\). This means the absolute value of the difference between x and y must be greater than 2.

Step 2 - Evaluate the First Pair (1, 3)

Calculate the absolute value of the difference between 1 and 3: \(|1-3| = |-2| = 2\). Since 2 is not greater than 2, \((1, 3)\) does not satisfy the inequality.

Step 3 - Evaluate the Second Pair (-2, 5)

Calculate the absolute value of the difference between -2 and 5: \(|-2-5| = |-7| = 7\). Since 7 is greater than 2, \((-2, 5)\) satisfies the inequality.

Step 4 - Evaluate the Third Pair (-6, -4)

Calculate the absolute value of the difference between -6 and -4: \(|-6-(-4)| = |-6+4| = |-2| = 2\). Since 2 is not greater than 2, \((-6, -4)\) does not satisfy the inequality.

Step 5 - Evaluate the Fourth Pair (7, -8)

Calculate the absolute value of the difference between 7 and -8: \(|7-(-8)| = |7+8| = |15| = 15\). Since 15 is greater than 2, \((7, -8)\) satisfies the inequality.

Key Concepts

Ordered Pairs Evaluation

When given a set of ordered pairs, it is important to understand how to evaluate them against a specific condition or inequality.
This involves calculating and comparing results using each pair.
In this case, we need to check if the inequality \(|x-y| > 2\) is satisfied for each pair.
To do this, we:

• Take each pair separately and label the first number as x and the second number as y.
• Apply the inequality condition to these values.
• Check if the resulting value satisfies the condition.

This process helps to determine if a pair meets the criteria set by the inequality, allowing for a clear and organized approach to problem-solving.

Absolute Value Calculation

The concept of absolute value is crucial in solving inequalities involving ordered pairs.
Absolute value refers to the distance of a number from zero, regardless of direction. It is always positive or zero.
Mathematically, for any number \(|a|\), if a is positive or zero, \(|a|= a\). If a is negative, \(|a|=-a\).

In our problem, we use this to calculate the difference between x and y for each pair:

• For the pair (1, 3): Calculate \(|1 - 3| = 2 \), since |-2| is 2.
• For the pair (-2, 5): Calculate \(|-2 - 5| = 7 \), since |-7| is 7.
• For the pair (-6, -4): Calculate \(|-6 - (-4)| = 2 \), since |-2| is 2.
• For the pair (7, -8): Calculate \(|7 - (-8)| = 15 \), since |15| is 15.

Using absolute values ensures that the distance is always considered positively, which is key to correctly solving these inequalities.

Compound Inequality Solution Steps

Solving compound inequalities like \(|x-y| > 2\) involves a systematic process:
1. Identify the operation required for each pair.
2. Perform the absolute value calculation.
3. Compare the calculated value with the condition given.
Let's break down the steps using examples:

• Step 1: Evaluate (1, 3): The calculation \(|1 - 3| = 2\) results in 2, which is not greater than 2. So (1, 3) does not satisfy the inequality.
• Step 2: Evaluate (-2, 5): The calculation \(|-2 - 5| = 7\) results in 7, which is greater than 2. Hence, (-2, 5) satisfies the inequality.
• Step 3: Evaluate (-6, -4): The calculation \(|-6 - (-4)| = 2\) results in 2, which is not greater than 2. So, (-6, -4) does not satisfy the inequality.
• Step 4: Evaluate (7, -8): The calculation \(|7 - (-8)| = 15\) results in 15, which is greater than 2. Hence, (7, -8) satisfies the inequality.

This methodical approach ensures a clear understanding and accurate comparison against the inequality.