Which of the following is the complete solution set to the inequality `|x|+|x-3|gt3` ?

Options

A. `{x : x gt 3 or x lt 0}`
B. `{x : -3 lt x lt 3}`
C. `{x : -3 gt x}`
D. `{x : -3 lt x}`

Explanation

To solve the inequality \( |x| + |x - 3| > 3 \), we will consider the different cases for the absolute values based on the critical points where the expressions inside the absolute values change sign. The critical points here are \( x = 0 \) and \( x = 3 \).

### Step 1: Identify the intervals
The critical points divide the number line into three intervals:
1. \( x < 0 \)
2. \( 0 \leq x < 3 \)
3. \( x \geq 3 \)

### Step 2: Solve for each interval

**Interval 1: \( x < 0 \)**
- Here, \( |x| = -x \) and \( |x - 3| = -(x - 3) = -x + 3 \).
- The inequality becomes:
\[ -x + (-x + 3) > 3 \]
Simplifying this:
\[ -2x + 3 > 3 \]
\[ -2x > 0 \]
\[ x < 0 \]
- This is valid for all \( x < 0 \).

**Interval 2: \( 0 \leq x < 3 \)**
- Here, \( |x| = x \) and \( |x - 3| = -(x - 3) = -x + 3 \).
- The inequality becomes:
\[ x + (-x + 3) > 3 \]
Simplifying this:
\[ 3 > 3 \]
- This is not valid, as it does not satisfy the inequality.

**Interval 3: \( x \geq 3 \)**
- Here, \( |x| = x \) and \( |x - 3| = x - 3 \).
- The inequality becomes:
\[ x + (x - 3) > 3 \]
Simplifying this:
\[ 2x - 3 > 3 \]
\[ 2x > 6 \]
\[ x > 3 \]
- This is valid for all \( x > 3 \).

### Step 3: Combine the solutions
From the intervals we analyzed:
- From Interval 1, we have \( x < 0 \).
- From Interval 2, there are no valid solutions.
- From Interval 3, we have \( x > 3 \).

Thus, the complete solution set is:
\[ x < 0 \quad \text{or} \quad x > 3 \]

### Final Answer
The complete solution set to the inequality \( |x| + |x - 3| > 3 \) is \( (-\infty, 0) \cup (3, \infty) \).

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