To solve the inequality \( x + 4 > 2\sqrt{4 - x^2} \), let's break it down step by step.

### Step 1: Consider the Domain of the Inequality
The square root function \( \sqrt{4 - x^2} \) requires that the expression inside the square root be non-negative. So, we need to find the values of \( x \) for which \( 4 - x^2 \geq 0 \).

\[
4 - x^2 \geq 0
\]
\[
x^2 \leq 4
\]
\[
-2 \leq x \leq 2
\]

So, the inequality is defined for \( x \) in the interval \( [-2, 2] \).

### Step 2: Solve the Inequality
Next, let's solve the inequality:

\[
x + 4 > 2\sqrt{4 - x^2}
\]

### Square Both Sides (If Possible)
Squaring both sides to eliminate the square root is a common technique, but it can introduce extraneous solutions, so we will need to check these solutions against the original inequality later.

\[
(x + 4)^2 > \left(2\sqrt{4 - x^2}\right)^2
\]
\[
x^2 + 8x + 16 > 4(4 - x^2)
\]
\[
x^2 + 8x + 16 > 16 - 4x^2
\]

### Step 3: Move All Terms to One Side
\[
x^2 + 8x + 16 + 4x^2 - 16 > 0
\]
\[
5x^2 + 8x > 0
\]

### Step 4: Factor the Quadratic Expression
\[
x(5x + 8) > 0
\]

This inequality will hold when the product of the two factors is positive. To solve \( x(5x + 8) > 0 \), consider the signs of the factors:

- \( x > 0 \) and \( 5x + 8 > 0 \)
- \( x < 0 \) and \( 5x + 8 < 0 \)

Let's analyze these cases:

1. **\( x > 0 \)**: Here, \( x \) is positive. For \( 5x + 8 > 0 \), solve:
   \[
   5x + 8 > 0
   \]
   \[
   5x > -8 \quad \Rightarrow \quad x > -\frac{8}{5}
   \]
   Since \( x > 0 \), the solution is \( x > 0 \).

2. **\( x < 0 \)**: Here, \( x \) is negative. For \( 5x + 8 < 0 \), solve:
   \[
   5x + 8 < 0
   \]
   \[
   5x < -8 \quad \Rightarrow \quad x < -\frac{8}{5}
   \]
   But remember that \( x \) must also satisfy \( -2 \leq x \leq 2 \), so \( x \) can only be in the interval \( [-2, -\frac{8}{5}] \).

### Step 5: Combine the Solutions and Check the Interval
So, the solution to the inequality is:
\[
x \in \left[-2, -\frac{8}{5}\right) \cup \left(0, 2\right]
\]

Finally, check these solutions in the original inequality to ensure no extraneous solutions have been introduced. This yields:

\[
x \in \left(-\frac{8}{5}, 2\right]
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you solve \( x^2 + 2x > 3 \)?
2. What is the solution to \( x + 4 \leq \sqrt{9 - x^2} \)?
3. How do you solve \( \sqrt{9 - x^2} = x + 1 \)?
4. What are the steps to solve \( x^2 - 4x + 3 < 0 \)?
5. How do you solve the inequality \( x^2 + x - 6 \geq 0 \)?

**Tip:** Always check the domain of a function before solving inequalities involving square roots or fractions.

Learn how to solve the inequality x + 4 > 2(sqrt(4-x^2)) with a detailed step-by-step guide. This problem covers key algebraic concepts, including inequalities involving square roots. Suitable for high school students in Grades 10-12.

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