Math Problem Statement

|x| + y = 4

Solution

The equation you've provided is:

x+y=4|x| + y = 4

To solve this equation, you would typically need to express it in terms of either xx or yy. Since it involves an absolute value, the solution can differ depending on the sign of xx.

Case 1: When x0x \geq 0

If xx is positive or zero, x=x|x| = x. The equation becomes:

x+y=4x + y = 4

This can be rearranged to solve for yy:

y=4xy = 4 - x

Case 2: When x<0x < 0

If xx is negative, x=x|x| = -x. The equation becomes:

x+y=4-x + y = 4

This can be rearranged to solve for yy:

y=4+xy = 4 + x

Final Answer

  • For x0x \geq 0, the equation is y=4xy = 4 - x.
  • For x<0x < 0, the equation is y=4+xy = 4 + x.

This represents two linear equations, with the solution depending on whether xx is non-negative or negative.

Would you like a graph of this, or any further clarification?


Here are five related questions you might find interesting:

  1. What is the graph of the equation x+y=4|x| + y = 4?
  2. How does the behavior of y=4xy = 4 - |x| compare to the linear equation without the absolute value?
  3. What happens to the equation if we add or subtract a constant from yy?
  4. How do you solve an equation involving absolute values for both xx and yy?
  5. What would the solution be if the equation were xy=4|x| - y = 4?

Tip: Always consider both cases when dealing with absolute value equations, as they often split the equation into two different forms depending on the sign of the variable inside the absolute value.

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Equations
Linear Equations

Formulas

Absolute value property: |x| = x if x ≥ 0, |x| = -x if x < 0
Equation rearrangement: y = 4 - x (for x ≥ 0), y = 4 + x (for x < 0)

Theorems

Absolute Value Properties
Linear Equation Theory

Suitable Grade Level

Grades 8-10