To solve the inequalities \( x^2 - 3x - 10 < 0 \) and \( 10x - x^2 - 16 > 0 \) simultaneously, we will break down the solution step by step.
### Step 1: Solve the first inequality \( x^2 - 3x - 10 < 0 \)
1. **Factor the quadratic expression**:
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]
Thus, we rewrite the inequality:
\[
(x - 5)(x + 2) < 0
\]
2. **Find the critical points**:
The critical points are \( x = 5 \) and \( x = -2 \).
3. **Test intervals**:
We will test the intervals determined by the critical points: \( (-\infty, -2) \), \( (-2, 5) \), and \( (5, \infty) \).
- For \( x < -2 \) (e.g., \( x = -3 \)):
\[
(-3 - 5)(-3 + 2) = (-8)(-1) = 8 > 0 \quad \text{(not valid)}
\]
- For \( -2 < x < 5 \) (e.g., \( x = 0 \)):
\[
(0 - 5)(0 + 2) = (-5)(2) = -10 < 0 \quad \text{(valid)}
\]
- For \( x > 5 \) (e.g., \( x = 6 \)):
\[
(6 - 5)(6 + 2) = (1)(8) = 8 > 0 \quad \text{(not valid)}
\]
4. **Conclusion for the first inequality**:
The solution for the first inequality is:
\[
-2 < x < 5
\]
### Step 2: Solve the second inequality \( 10x - x^2 - 16 > 0 \)
1. **Rearrange the inequality**:
\[
-x^2 + 10x - 16 > 0
\]
Multiply through by -1 (reversing the inequality):
\[
x^2 - 10x + 16 < 0
\]
2. **Factor the quadratic expression**:
\[
x^2 - 10x + 16 = (x - 2)(x - 8)
\]
Thus, we rewrite the inequality:
\[
(x - 2)(x - 8) < 0
\]
3. **Find the critical points**:
The critical points are \( x = 2 \) and \( x = 8 \).
4. **Test intervals**:
We will test the intervals determined by the critical points: \( (-\infty, 2) \), \( (2, 8) \), and \( (8, \infty) \).
- For \( x < 2 \) (e.g., \( x = 0 \)):
\[
(0 - 2)(0 - 8) = (-2)(-8) = 16 > 0 \quad \text{(not valid)}
\]
- For \( 2 < x < 8 \) (e.g., \( x = 4 \)):
\[
(4 - 2)(4 - 8) = (2)(-4) = -8 < 0 \quad \text{(valid)}
\]
- For \( x > 8 \) (e.g., \( x = 9 \)):
\[
(9 - 2)(9 - 8) = (7)(1) = 7 > 0 \quad \text{(not valid)}
\]
5. **Conclusion for the second inequality**:
The solution for the second inequality is:
\[
2 < x < 8
\]
### Step 3: Find the intersection of the two solutions
1. **First inequality**: \( -2 < x < 5 \)
2. **Second inequality**: \( 2 < x < 8 \)
The intersection of these two intervals is:
\[
2 < x < 5
\]
### Final Answer:
The set of values of \( x \) for which both inequalities hold simultaneously is:
\[
(2, 5)
\]
