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.