Math Problem Statement

Simplify the expression \(\frac{\|x - 1\| - \|-2y\|}{\|3y + 2x\|}\).

Solution

The expression you uploaded is:

x12y3y+2x\frac{|x - 1| - |-2y|}{|3y + 2x|}

This expression involves absolute values. Let's break it down:

Numerator:

  • x1|x - 1| is the absolute value of x1x - 1.
  • 2y=2y|-2y| = 2|y| because the absolute value of 2y-2y is 2×y2 \times |y|.

So the numerator simplifies to: x12y|x - 1| - 2|y|

Denominator:

  • 3y+2x|3y + 2x| remains as is because it is the absolute value of the linear combination 3y+2x3y + 2x.

Thus, the entire expression simplifies to:

x12y3y+2x\frac{|x - 1| - 2|y|}{|3y + 2x|}

Let me know if you need further simplifications or details, and feel free to ask any clarifying questions!


Here are five related questions that might expand on this:

  1. How do you solve equations involving absolute values?
  2. What is the geometric interpretation of absolute value functions?
  3. How do we handle cases when the expression inside an absolute value equals zero?
  4. Can you graph this function to understand its behavior?
  5. How do absolute values affect the domain and range of this expression?

Tip: When simplifying absolute value expressions, consider the sign of the terms inside the absolute values, as they could affect the overall result!

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Math Problem Analysis

Mathematical Concepts

Absolute Values
Linear Expressions
Simplification

Formulas

|a|, |b| for absolute values
|-a| = |a|

Theorems

Properties of Absolute Value

Suitable Grade Level

Grades 8-10