1. Given inequality $$ -\frac{6t}{t^2-2} $$$$ <-\frac{30}{t^2-2} $$

a) Solve it using an algebraic strategy. [4 marks]

    b) Support your answer in a) with a geometric solution. 

Let's solve the inequality step by step.

### a) Solve the inequality algebraically

Given inequality:

\[
-\frac{6t}{t^2 - 2} < -\frac{30}{t^2 - 2}
\]

1. **Multiply both sides by $-(t^2 - 2)$:**

   Since we are multiplying by a negative expression, the inequality sign will flip.

   \[
   6t > 30
   \]

2. **Solve for $ t $:**

   Divide both sides by 6:

   \[
   t > 5
   \]

3. **Consider the domain of the original inequality:**

   The expression $ t^2 - 2 $ should not be zero, so:

   \[
   t^2 - 2 \neq 0 \implies t^2 \neq 2 \implies t \neq \pm \sqrt{2}
   \]

   Additionally, $ t^2 - 2 > 0 $ to keep the denominators positive (except for the negative sign in front), which means:

   \[
   t^2 > 2 \implies t > \sqrt{2} \text{ or } t < -\sqrt{2}
   \]

4. **Combine the results:**

   Considering both the solution $ t > 5 $ and the domain restrictions $ t > \sqrt{2} $ or $ t < -\sqrt{2} $, the valid solution is:

   \[
   t > 5
   \]

### b) Geometric Solution

To support the algebraic solution geometrically, we can analyze the graphs of the two functions involved in the inequality:

1. **Functions to graph:**

   \[
   f(t) = -\frac{6t}{t^2 - 2} \quad \text{and} \quad g(t) = -\frac{30}{t^2 - 2}
   \]

2. **Intersection and Comparison:**

   - Graph $ f(t) = -\frac{6t}{t^2 - 2} $: This is a hyperbola with asymptotes at $ t = \pm \sqrt{2} $.
   - Graph $ g(t) = -\frac{30}{t^2 - 2} $: This is another hyperbola with the same asymptotes.

   The inequality $ -\frac{6t}{t^2 - 2} < -\frac{30}{t^2 - 2} $ translates to finding where the graph of $ f(t) $ is below the graph of $ g(t) $.

3. **Interpret the intersection points:**

   Solve $ f(t) = g(t) $ to find intersection points, which gives $ t = 5 $.

4. **Analyze the regions:**

   The function $ f(t) $ will be less than $ g(t) $ when $ t > 5 $, which aligns with our algebraic solution.

This geometric approach confirms that the solution $ t > 5 $ is correct.

---

Would you like further details or have any questions about this explanation? Here are some related questions:

1. What are the steps to graph a rational function?
2. How do you determine the asymptotes of a hyperbola?
3. What are the properties of hyperbolas with respect to inequalities?
4. Can you explain how multiplying an inequality by a negative value affects the inequality sign?
5. What are domain restrictions in rational inequalities?
6. How do intersections of graphs relate to solving inequalities?
7. What other methods can be used to solve rational inequalities?
8. How do you verify solutions of inequalities algebraically?

**Tip:** Always check the domain when solving inequalities involving rational expressions to avoid invalid solutions.

Learn how to solve the rational inequality -6t/(t^2-2) < -30/(t^2-2) algebraically and geometrically. This problem involves inequalities, rational functions, and domain restrictions. Suitable for advanced high school students (Grades 10-12).

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