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Magazine Chapter 2: Problem 74
Assume \(A>B .\) Is it always, sometimes, or never true that \(A-|B| \leq A+B ?\) Explain,
Short Answer
Expert verified
The inequality is always true.
Step by step solution
01
Understand the Comparison
We're given that \( A > B \). This comparison concerns the size of \( A \) relative to \( B \). Since \( A \) is greater than \( B \), the positive difference \( A - B \) is meaningful.
02
Understand the Absolute Value
The expression \(|B|\) represents the absolute value of \(B\), which means \(|B| = B\) if \( B \geq 0 \), and \(|B| = -B\) if \( B < 0 \).
03
Analyze the Expression A - |B|
The expression \( A - |B| \) can take different values depending on whether \( B \) is positive or negative. If \( B \geq 0 \), then \( A - |B| = A - B \). If \( B < 0 \), then \( A - |B| = A + B \).
04
Compare A - |B| to A + B for B 鈮� 0
When \( B \geq 0 \), we have \( A - |B| = A - B \) and \( A + B \). Hence, \( A - B \) is always less than or equal to \( A + B \) as long as both sides are defined, since \( B \geq 0 \).
05
Compare A - |B| to A + B for B < 0
When \( B < 0 \), we have \( A - |B| = A + B \). In this case, the expression \( A - |B| \) is exactly equal to \( A + B \).
06
Conclusion
In both cases (\( B \geq 0 \) and \( B < 0 \)), the inequality \( A - |B| \leq A + B \) is true, as either it becomes an equality or \( A - B \) is less than or equal to \( A + B \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Absolute Value
Absolute value is a concept that simplifies expressions by focusing on magnitude rather than direction. It is denoted by vertical bars. For example, the absolute value of a number \(B\) is written as \(|B|\). It represents the distance of \(B\) from zero on a number line, regardless of direction.
- If \(B\) is positive or zero, \(|B| = B\).
- If \(B\) is negative, \(|B| = -B\).
Expressions Involving Variables
Expressions that contain variables, like \(A-|B|\) and \(A+B\), require careful handling, especially when considering inequalities. Variables can represent a wide range of values, which requires us to analyze certain parts of an expression based on conditions given, such as \(B \geq 0\) or \(B < 0\).
Understanding how the variables interact with operators (like addition or subtraction) and functions (like absolute value) is essential. For example:
Understanding how the variables interact with operators (like addition or subtraction) and functions (like absolute value) is essential. For example:
- When \(B\) is positive, \(A-|B|\) can be directly simplified to \(A-B\).
- When \(B\) is negative, \(A-|B|\) transforms into \(A+B\).
Strategies for Problem Solving with Inequalities
In problem solving, especially with inequalities involving absolute values and variables, a systematic approach is beneficial. Here are some strategies to keep in mind:
- Start by clearly understanding the problem: Identify the variables, constants, and any conditions such as \(A>B\).
- Break down complex expressions into simpler parts. In this problem, analyze what happens in both scenarios when \(B \geq 0\) and when \(B < 0\).
- Compare parts of the expressions separately. This helps in understanding how changes in the value of \(B\) affect the overall expression \(A-|B|\).
- Conclude by verifying if the inequality holds in all cases, as with our conclusion that \(A-|B|\leq A+B\) is always true whether \(B\) is positive or negative.
