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To solve the problem of finding how the expression \( 2x - x|x - 1| + x|x| + 3 \) looks on the interval \( (-\infty, 0) \) after removing the absolute values, we will analyze each term in the expression.

### Step 1: Understand the absolute value conditions
Since the interval is \( (-\infty, 0) \), we know that:
- \( x < 0 \)
- \( x - 1 < 0 \) because \( x < 0 \).

### Step 2: Rewrite the absolute values based on the interval
Using the conditions, we can simplify the absolute values as follows:
- \( |x - 1| = -(x - 1) = -x + 1 \) because \( x - 1 < 0 \).
- \( |x| = -x \) because \( x < 0 \).

### Step 3: Substitute these values back into the expression
The original expression is:
\[ 2x - x|x - 1| + x|x| + 3 \]

Substituting the values of \( |x - 1| \) and \( |x| \) we get:
\[ 2x - x(-x + 1) + x(-x) + 3 \]

### Step 4: Simplify the expression
Expanding each term:
\[ 2x - (-x^2 + x) - x^2 + 3 \]
\[ 2x + x^2 - x - x^2 + 3 \]

Combine like terms:
\[ 2x - x + 3 = x + 3 \]

### Step 5: Check the answer options
The correct simplified form of the expression on the interval \( (-\infty, 0) \) is \( x + 3 \).

### Answer
The correct answer is:
\[ x + 3 \]

Would you like more details or have any questions?

### Related Questions:
1. How do you deal with absolute values when the interval changes to \( (0, \infty) \)?
2. Can absolute values ever be treated differently on other types of intervals?
3. What is the geometric interpretation of absolute value transformations?
4. How do we approach simplification when multiple absolute values are involved?
5. How do we handle expressions involving absolute values with variables that have unknown signs?

### Tip:
When dealing with absolute values, always check the sign of the expression inside the absolute value on the specified interval before simplifying.

Vyberte ze seznamu, jak bude vypadat výraz 2x - x|x - 1| + x|x| + 3 na intervalu (-∞, 0) po odstranění absolutní hodnoty:

Learn how to simplify the expression 2x - x|x - 1| + x|x| + 3 on the interval (-∞, 0) by removing absolute values and simplifying terms. This step-by-step guide is suitable for students in Grades 9-10 and covers key algebra concepts like handling absolute values and simplifying linear expressions.

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