Math Problem Statement

حل کن

If |a - b| < |a| + |b| and (|a| / a) + (|b| / b) = -3, what is the result?

Solution

The problem in the image asks for a solution to an inequality involving absolute values. Here's the breakdown:

Given:

ab<a+b|a - b| < |a| + |b| and aa+bb=3\frac{|a|}{a} + \frac{|b|}{b} = -3

You are asked to find which value corresponds to this condition among the following options:

  1. 11
  2. 00
  3. 2-2
  4. 1-1

Step 1: Analyze the Inequality

The inequality ab<a+b|a - b| < |a| + |b| is always true due to the triangle inequality for real numbers. This condition does not provide additional restrictions on the values of aa and bb.

Step 2: Solve the Fractional Equation

The equation aa+bb=3\frac{|a|}{a} + \frac{|b|}{b} = -3 is key here. This expression involves the sign function:

  • aa\frac{|a|}{a} equals:
    • 11 if a>0a > 0
    • 1-1 if a<0a < 0
    • Undefined if a=0a = 0 (but we'll assume a0a \neq 0)

Similarly, for bb\frac{|b|}{b}, it is 11 if b>0b > 0 and 1-1 if b<0b < 0.

To satisfy the equation aa+bb=3\frac{|a|}{a} + \frac{|b|}{b} = -3, both aa and bb must be negative.

  • aa=1\frac{|a|}{a} = -1 (since a<0a < 0)
  • bb=1\frac{|b|}{b} = -1 (since b<0b < 0)

Thus: 1+(1)=2-1 + (-1) = -2 This suggests that the correct answer is 2-2.

Final Answer:

The solution to the problem is option 3: 2-2.


Do you have any questions, or would you like more details?

Here are 5 related questions you might find interesting:

  1. What is the significance of the triangle inequality in real numbers?
  2. How does the sign function xx\frac{|x|}{x} behave for different values of xx?
  3. Why is aa+bb=3\frac{|a|}{a} + \frac{|b|}{b} = -3 not possible for positive values of aa and bb?
  4. How would the solution change if the equation were aa+bb=0\frac{|a|}{a} + \frac{|b|}{b} = 0?
  5. Can absolute value equations have more than one solution in some cases?

Tip: When dealing with absolute values, it's often helpful to break the problem into cases based on the sign of the variables.

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Math Problem Analysis

Mathematical Concepts

Absolute Value
Inequalities
Sign Function

Formulas

Triangle Inequality: |a - b| ≤ |a| + |b|
Sign Function: (|x| / x) = 1 if x > 0, -1 if x < 0

Theorems

Triangle Inequality

Suitable Grade Level

Grades 10-12