To solve the inequality \(\frac{2t - 7}{t - 4} \geq 3\), follow these steps:

### Step 1: Set the inequality as an equation
First, treat the inequality as an equation to help find critical points:
\[
\frac{2t - 7}{t - 4} = 3
\]

### Step 2: Clear the fraction by multiplying both sides by \((t - 4)\)
Assuming \(t - 4 \neq 0\) (so we avoid dividing by zero), multiply both sides by \((t - 4)\):
\[
2t - 7 = 3(t - 4)
\]

### Step 3: Expand and simplify the equation
Distribute on the right-hand side:
\[
2t - 7 = 3t - 12
\]
Next, move all terms involving \(t\) to one side:
\[
2t - 3t = -12 + 7
\]
Simplify:
\[
-t = -5
\]
Now, divide both sides by \(-1\):
\[
t = 5
\]
This is one of the critical points. However, remember that solving the equation is just one part; next, we need to analyze the inequality.

### Step 4: Find the sign of the inequality around critical points
The inequality \(\frac{2t - 7}{t - 4} \geq 3\) changes sign at \(t = 5\) and at the point where the denominator is zero, \(t = 4\). These points divide the number line into intervals. Check the sign of the inequality within these intervals:

1. **Interval 1**: \(t < 4\)
2. **Interval 2**: \(4 < t < 5\)
3. **Interval 3**: \(t > 5\)

### Step 5: Test the intervals
For each interval, pick a test point and check if the inequality holds:

- **For \(t < 4\)** (e.g., \(t = 0\)):
  \[
  \frac{2(0) - 7}{0 - 4} = \frac{-7}{-4} = \frac{7}{4} < 3
  \]
  This interval does not satisfy the inequality.

- **For \(4 < t < 5\)** (e.g., \(t = 4.5\)):
  \[
  \frac{2(4.5) - 7}{4.5 - 4} = \frac{9 - 7}{0.5} = \frac{2}{0.5} = 4 > 3
  \]
  This interval satisfies the inequality.

- **For \(t > 5\)** (e.g., \(t = 6\)):
  \[
  \frac{2(6) - 7}{6 - 4} = \frac{12 - 7}{2} = \frac{5}{2} = 2.5 < 3
  \]
  This interval does not satisfy the inequality.

### Step 6: Include boundary points
- At \(t = 5\), the original inequality becomes \(3 \geq 3\), which is true.
- At \(t = 4\), the expression is undefined, so \(t = 4\) is not included in the solution.

### Final Answer
The solution is:
\[
4 < t \leq 5
\]
This represents the values of \(t\) that satisfy the inequality.

Would you like more details on any of these steps?

### Relative Questions:
1. How would the solution change if the inequality was \(\frac{2t - 7}{t - 4} > 3\)?
2. What happens if the inequality is reversed to \(\frac{2t - 7}{t - 4} \leq 3\)?
3. How do you solve similar inequalities where the expression is quadratic?
4. How do you check for extraneous solutions in rational inequalities?
5. Can this method be applied to systems of inequalities?

### Tip:
When dealing with rational inequalities, always remember to check for points where the denominator equals zero, as these points are excluded from the solution set.

Explore the step-by-step process to solve the rational inequality (2t-7)/(t-4) ≥ 3. Learn about critical points, sign analysis, and how to handle undefined expressions in rational inequalities. This solution is ideal for high school students in Grades 9-12.

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