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Let's solve each part of the inequality problem step-by-step.

### Problem (a): Solve \( x^2 + 8x - 9 \leq 0 \)

1. **Factor the Quadratic Expression:**
   \[
   x^2 + 8x - 9 = (x + 9)(x - 1)
   \]

2. **Set Each Factor to Zero to Find Critical Points:**
   \[
   x + 9 = 0 \Rightarrow x = -9
   \]
   \[
   x - 1 = 0 \Rightarrow x = 1
   \]

3. **Determine the Intervals to Test:**
   The critical points divide the number line into intervals:
   - \( (-\infty, -9) \)
   - \( (-9, 1) \)
   - \( (1, \infty) \)

4. **Test Each Interval in the Inequality \( (x + 9)(x - 1) \leq 0 \):**
   - For \( x \in (-\infty, -9) \), pick \( x = -10 \): \((x + 9)(x - 1) > 0\).
   - For \( x \in (-9, 1) \), pick \( x = 0 \): \((x + 9)(x - 1) < 0\).
   - For \( x \in (1, \infty) \), pick \( x = 2 \): \((x + 9)(x - 1) > 0\).

   The inequality \( (x + 9)(x - 1) \leq 0 \) holds for the interval \( x \in [-9, 1] \).

5. **Answer for (a):**
   \[
   [-9, 1]
   \]

---

### Problem (b): Solve \( 10x^2 + x + 9 > 0 \)

1. **Determine if the Quadratic Can Be Factored Easily:**
   The expression \( 10x^2 + x + 9 \) does not factor easily, so we use the discriminant to determine if it has real roots.

2. **Calculate the Discriminant:**
   For a quadratic \( ax^2 + bx + c \), the discriminant is \( b^2 - 4ac \).
   Here, \( a = 10 \), \( b = 1 \), and \( c = 9 \).
   \[
   \text{Discriminant} = 1^2 - 4 \cdot 10 \cdot 9 = 1 - 360 = -359
   \]
   Since the discriminant is negative, there are no real roots, meaning the quadratic does not cross the x-axis.

3. **Determine the Sign of \( 10x^2 + x + 9 \):**
   Since the coefficient of \( x^2 \) (10) is positive and there are no real roots, \( 10x^2 + x + 9 \) is always positive for all \( x \in \mathbb{R} \).

4. **Answer for (b):**
   \[
   (-\infty, \infty)
   \]

---

### Summary of Answers:
- **(a)** \( x^2 + 8x - 9 \leq 0 \): **Answer**: \([-9, 1]\)
- **(b)** \( 10x^2 + x + 9 > 0 \): **Answer**: \((-\infty, \infty)\)

Would you like further explanations or have any questions on the steps taken?

---

### Additional Practice Questions
1. How would the solution change if the inequality in (a) was \( x^2 + 8x - 9 < 0 \)?
2. Can you solve \( x^2 + 8x - 9 \geq 0 \)? What would the answer be in interval notation?
3. What is the impact of a negative discriminant on a quadratic inequality?
4. How does changing the sign of the leading coefficient in a quadratic equation affect its inequality solution?
5. Can you graph \( y = x^2 + 8x - 9 \) and \( y = 10x^2 + x + 9 \) to visualize the solutions?

### Tip
For quadratic inequalities, identifying critical points and testing intervals is crucial to determining the solution set.

Solve the following inequalities. Enter the answers in interval notation.
(a) x^2 + 8x - 9 ≤ 0
(b) 10x^2 + x + 9 > 0

Learn how to solve quadratic inequalities such as x^2 + 8x - 9 ≤ 0 and 10x^2 + x + 9 > 0. This problem explores factorization, interval testing, and the discriminant method for inequalities in interval notation, suitable for students in Grades 10-12.

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