What is    16 - y|x + y| + y| x|- y2        if   y< x < 0 ?



We are given the expression:

\[
16 - y|x + y| + y|x| - y^2
\]
with the condition \( y < x < 0 \).

Let's break this down step by step:

### Step 1: Analyze the absolute values
Given \( y < x < 0 \), both \( x \) and \( y \) are negative. Therefore:
- \( |x + y| = -(x + y) \) because \( x + y < 0 \),
- \( |x| = -x \) because \( x < 0 \).

### Step 2: Rewrite the expression
Now substitute the absolute value terms back into the expression:
\[
16 - y|x + y| + y|x| - y^2 = 16 - y(-x - y) + y(-x) - y^2
\]
This simplifies to:
\[
16 - y(-x - y) + y(-x) - y^2 = 16 + y(x + y) - yx - y^2
\]

### Step 3: Simplify the expression
Expand the terms:
\[
= 16 + yx + y^2 - yx - y^2
\]
The terms \( yx \) and \( -yx \), as well as \( y^2 \) and \( -y^2 \), cancel each other out. Thus, we are left with:
\[
= 16
\]

### Final Answer:
The value of the expression is \( 16 \).

Would you like further explanation, or do you have any questions? Here are some related questions you might find helpful:

1. What happens if \( y > x \)?
2. How do absolute values affect expressions when both variables are negative?
3. How would the result change if \( y = x \)?
4. What is the effect of flipping the inequality \( y < x < 0 \)?
5. Can you solve similar problems with different conditions on \( x \) and \( y \)?

**Tip:** Always consider the sign of variables inside absolute values, especially when working with inequalities.

What is 16 - y|x + y| + y|x| - y^2 if y < x < 0?

Simplify the algebraic expression 16 - y|x + y| + y|x| - y^2 with the condition y < x < 0. This problem explores key algebraic concepts like absolute value properties and inequalities, making it ideal for high school students in Grades 9-12.

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