Solve `(x+3)^(5)-(x-1)^(5)ge244`.

Answer

To solve the inequality \((x+3)^{5} - (x-1)^{5} \geq 244\), we can follow these steps:

### Step 1: Rewrite the inequality
We start with the inequality:
\[ (x+3)^{5} - (x-1)^{5} \geq 244 \]

### Step 2: Evaluate at a specific point
Let’s check the value of the expression when \(x = 0\):
\[ (0+3)^{5} - (0-1)^{5} = 3^{5} - (-1)^{5} \]
Calculating each term:
\[ 3^{5} = 243 \quad \text{and} \quad (-1)^{5} = -1 \]
Thus, we have:
\[ 243 - (-1) = 243 + 1 = 244 \]
This means when \(x = 0\), the left-hand side equals 244.

### Step 3: Determine the behavior of the function
Next, we need to analyze how the expression behaves as \(x\) increases. The function \((x+3)^{5}\) and \((x-1)^{5}\) are both increasing functions because they are polynomial functions with positive leading coefficients.

### Step 4: Consider values greater than 0
Since we found that at \(x = 0\), the left-hand side equals 244, we now need to check if the left-hand side is greater than 244 for \(x > 0\):
- For any \(x > 0\), both \((x+3)^{5}\) and \((x-1)^{5}\) will increase, hence \((x+3)^{5} - (x-1)^{5}\) will also increase.

### Step 5: Conclusion
Since we found that the left-hand side equals 244 at \(x = 0\) and increases for \(x > 0\), we conclude that:
\[ (x+3)^{5} - (x-1)^{5} \geq 244 \quad \text{for} \quad x \geq 0 \]

Thus, the solution to the inequality is:
\[ x \in [0, \infty) \]