Question 36 , solve the inequalities 3$$ \frac{x}{x-4} $$$$ \ge\frac{2x}{7} $$

To solve the inequality $ \frac{3x}{x-4} \geq \frac{2x}{7} $, follow these steps:

### Step 1: Eliminate the fractions
Multiply both sides by $ 7(x-4) $ to eliminate the denominators. Note that $ x \neq 4 $ because that would make the denominator zero. This multiplication gives:
\[
7(x-4) \times \frac{3x}{x-4} \geq 7(x-4) \times \frac{2x}{7}
\]

Simplify the expression:
\[
7 \cdot 3x \geq 2x(x-4)
\]

This simplifies to:
\[
21x \geq 2x^2 - 8x
\]

### Step 2: Rearrange the inequality
Bring all terms to one side of the inequality:
\[
0 \geq 2x^2 - 8x - 21x
\]

Simplify further:
\[
0 \geq 2x^2 - 29x
\]

Or:
\[
2x^2 - 29x \leq 0
\]

### Step 3: Factor the quadratic expression
Factor out $ x $:
\[
x(2x - 29) \leq 0
\]

### Step 4: Determine the critical points
The critical points are where the expression equals zero:
\[
x = 0 \quad \text{or} \quad 2x - 29 = 0 \Rightarrow x = \frac{29}{2} = 14.5
\]

### Step 5: Test intervals
To determine where the inequality holds, test intervals around the critical points $ x = 0 $ and $ x = 14.5 $:

1. **Interval $ (-\infty, 0) $**: 
   - Choose $ x = -1 $:
   \[
   x(2x - 29) = (-1)(2(-1) - 29) = -1(-2 - 29) = -1(-31) = 31 > 0
   \]
   - This interval is not a solution.

2. **Interval $ (0, 14.5) $**: 
   - Choose $ x = 1 $:
   \[
   x(2x - 29) = (1)(2(1) - 29) = 1(2 - 29) = 1(-27) = -27 < 0
   \]
   - This interval is a solution.

3. **Interval $ (14.5, \infty) $**:
   - Choose $ x = 15 $:
   \[
   x(2x - 29) = (15)(2(15) - 29) = 15(30 - 29) = 15(1) = 15 > 0
   \]
   - This interval is not a solution.

### Step 6: Conclusion
The solution for the inequality is $ x \in (0, 14.5] $.

This includes all values of $ x $ where the expression is less than or equal to zero, excluding $ x = 4 $, as it makes the original denominator zero.

---

Would you like more details on any step? Or, do you have any other questions? Here are some related questions:

### 8 Related Questions:
1. How can you determine the domain of a rational function?
2. What are the steps to solve a quadratic inequality?
3. How do you graph the solution set of an inequality?
4. How do critical points affect the solution of inequalities?
5. How do you check for extraneous solutions in inequalities?
6. What techniques are used to factor quadratic expressions?
7. How do you solve rational equations involving multiple fractions?
8. What is the significance of sign analysis in solving inequalities?

### Tip:
When multiplying or dividing by an expression containing a variable in inequalities, be mindful of the sign, as it can affect the direction of the inequality. Always consider the domain restrictions that might arise from these operations.

Learn how to solve the rational inequality (3x)/(x-4) >= (2x)/7 step-by-step with detailed calculations and interval testing. This problem involves inequalities, rational functions, and quadratic equations, suitable for advanced high school students (Grades 10-12).

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