Solve `|x-pi|+|x^2-pi^2| le 0 `

Explanation

To solve the inequality \( |x - \pi| + |x^2 - \pi^2| \leq 0 \), we will follow these steps:

### Step 1: Understand the properties of absolute values
The absolute value \( |a| \) is always non-negative, meaning \( |a| \geq 0 \) for any real number \( a \). Therefore, both \( |x - \pi| \) and \( |x^2 - \pi^2| \) are non-negative.

### Step 2: Analyze the inequality
Since both terms in the inequality are non-negative, the only way their sum can be less than or equal to zero is if both terms are equal to zero:
\[ |x - \pi| = 0 \quad \text{and} \quad |x^2 - \pi^2| = 0 \]

### Step 3: Solve the first equation
From \( |x - \pi| = 0 \), we have:
\[ x - \pi = 0 \implies x = \pi \]

### Step 4: Solve the second equation
From \( |x^2 - \pi^2| = 0 \), we have:
\[ x^2 - \pi^2 = 0 \implies x^2 = \pi^2 \]
This gives us two solutions:
\[ x = \pi \quad \text{or} \quad x = -\pi \]

### Step 5: Combine the results
To satisfy the original inequality \( |x - \pi| + |x^2 - \pi^2| \leq 0 \), both conditions must hold true. Since \( |x - \pi| = 0 \) gives \( x = \pi \), we check if this value satisfies the second condition:
- For \( x = \pi \):
\[ |x^2 - \pi^2| = | \pi^2 - \pi^2 | = 0 \]
Thus, \( x = \pi \) satisfies both conditions.

- For \( x = -\pi \):
\[ |x - \pi| = | -\pi - \pi | = | -2\pi | = 2\pi > 0 \]
This does not satisfy the first condition.

### Conclusion
The only solution to the inequality \( |x - \pi| + |x^2 - \pi^2| \leq 0 \) is:
\[ \boxed{\pi} \]