https://file.aiquickdraw.com/mathbot/file/1725841566437x2ueqxjk.jpg

The inequality given in the image is:

\[
\frac{4t + 5}{t} > 3
\]

To solve this inequality, let's follow these steps:

1. **Multiply both sides by \( t \)** (assuming \( t \neq 0 \)) to eliminate the fraction:
   
   \[
   4t + 5 > 3t
   \]

2. **Subtract \( 3t \) from both sides** to isolate the variable on one side:

   \[
   4t - 3t + 5 > 0
   \]

   Simplifying this:

   \[
   t + 5 > 0
   \]

3. **Subtract 5 from both sides** to solve for \( t \):

   \[
   t > -5
   \]

4. **Consider the case when \( t = 0 \)** since we multiplied by \( t \) initially. The original expression \( \frac{4t + 5}{t} \) is undefined when \( t = 0 \), so \( t \neq 0 \).

Thus, the solution to the inequality is:

\[
t > -5 \quad \text{and} \quad t \neq 0
\]

This means \( t \) can be any value greater than \(-5\), except \( t \) cannot be 0.

### Do you want more details, or do you have any questions?

### Related Questions:

1. What happens if \( t \) is negative? How does it affect the inequality?
2. How would the solution change if the inequality was \( \geq 3 \) instead of \( > 3 \)?
3. Can you solve similar inequalities with different constants, like \( \frac{2t + 7}{t} > 4 \)?
4. What would the solution be if the inequality was \( \frac{4t + 5}{t} < 3 \) instead?
5. How would you solve \( \frac{4t + 5}{t} = 3 \)?

### Tip:

Always check for cases where a variable might make the denominator zero, as this can affect the validity of the solution.

Learn how to solve the inequality (4t + 5)/t > 3 by multiplying both sides by t, simplifying the expression, and considering the undefined case where t = 0. This problem covers key algebraic concepts related to inequalities and rational expressions. Suitable for high school students in Grades 10-12.

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