$$  $$Given inequality $$ -\frac{6t}{t^2-2} $$$$ <-\frac{30}{t^2-2} $$a) Solve it using an algebraic strategy.

To solve the inequality:

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

### Step 1: Analyze the inequality and possible restrictions

The given inequality involves rational expressions. The denominator $ t^2 - 2 $ cannot be zero, so we need to determine when $ t^2 - 2 \neq 0 $:

\[
t^2 - 2 = 0
\]

Solving for $ t $:

\[
t = \pm\sqrt{2}
\]

Thus, $ t \neq \sqrt{2} $ and $ t \neq -\sqrt{2} $.

### Step 2: Multiply both sides by $ -(t^2 - 2) $

Since $ t^2 - 2 $ is positive for $ t \in (-\sqrt{2}, \sqrt{2}) $ and negative for $ t \notin (-\sqrt{2}, \sqrt{2}) $, multiplying by $ -(t^2 - 2) $ will reverse the inequality sign. So we get:

\[
6t > 30
\]

### Step 3: Solve the resulting inequality

Now, solve for $ t $:

\[
6t > 30
\]

Divide both sides by 6:

\[
t > 5
\]

### Step 4: Consider the restrictions

Since the denominator $ t^2 - 2 $ must not be zero, $ t $ cannot be $ \pm \sqrt{2} $. Additionally, we need to consider where $ t^2 - 2 $ is positive and negative.

The expression $ t^2 - 2 $ is positive for $ t > \sqrt{2} $ or $ t < -\sqrt{2} $, and negative for $ -\sqrt{2} < t < \sqrt{2} $.

Given the condition $ t > 5 $ from the inequality, this is within the region where $ t^2 - 2 $ is positive.

### Final Answer

The solution to the inequality is:

\[
t > 5
\]

Would you like further details or have any questions?

### Related Questions

1. How would the solution change if the inequality were $ \frac{6t}{t^2 - 2} > \frac{30}{t^2 - 2} $?
2. What happens if the inequality had $ t^2 - 4 $ in the denominator instead of $ t^2 - 2 $?
3. How does the solution change if $ t^2 - 2 $ is replaced by another quadratic expression?
4. What are the implications of multiplying or dividing both sides of the inequality by a negative quantity?
5. How would you solve the inequality if $ t^2 - 2 $ were a cubic expression instead of quadratic?
6. What if the inequality were $ -\frac{6t}{t^2 - 2} \leq -\frac{30}{t^2 - 2} $?
7. How does the solution change if the inequality involves absolute values, such as $ -\frac{6t}{|t^2 - 2|} < -\frac{30}{|t^2 - 2|} $?
8. How does the presence of a quadratic in the denominator affect the possible solutions for the inequality?

### Tip

When solving inequalities involving rational expressions, be careful with the signs of the expressions and remember that multiplying or dividing by a negative quantity reverses the inequality sign.

Learn how to solve the rational inequality -6t / (t^2 - 2) < -30 / (t^2 - 2) step-by-step with detailed explanations. This problem involves inequalities and rational expressions, suitable for high school students.

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