The values of x for which `(x-1)/(3x+4) lt (x-3)/(3x-2)`

Options

A. `(-oo, (5)/(4))`
B. `(-(4)/(3), (4)/(3))`
C. `((3)/(4), oo)`
D. `(-oo, -(5)/(4)) uu (3//4, oo)`

Answer

To solve the inequality \(\frac{x-1}{3x+4} < \frac{x-3}{3x-2}\), we will follow these steps:

### Step 1: Rewrite the Inequality
Start by rewriting the inequality:
\[ \frac{x-1}{3x+4} - \frac{x-3}{3x-2} < 0 \]

### Step 2: Find a Common Denominator
The common denominator for the fractions is \((3x + 4)(3x - 2)\). Rewrite the inequality with this common denominator:
\[ \frac{(x-1)(3x-2) - (x-3)(3x+4)}{(3x+4)(3x-2)} < 0 \]

### Step 3: Expand the Numerator
Now, expand the numerator:
1. For \((x-1)(3x-2)\):
\[ x(3x) - x(2) - 1(3x) + 1(2) = 3x^2 - 2x - 3x + 2 = 3x^2 - 5x + 2 \]

2. For \((x-3)(3x+4)\):
\[ x(3x) + x(4) - 3(3x) - 3(4) = 3x^2 + 4x - 9x - 12 = 3x^2 - 5x - 12 \]

Now, substitute these back into the inequality:
\[ \frac{(3x^2 - 5x + 2) - (3x^2 - 5x - 12)}{(3x+4)(3x-2)} < 0 \]

### Step 4: Simplify the Numerator
Simplify the numerator:
\[ 3x^2 - 5x + 2 - 3x^2 + 5x + 12 = 14 \]
So, the inequality becomes:
\[ \frac{14}{(3x+4)(3x-2)} < 0 \]

### Step 5: Analyze the Sign of the Expression
The fraction \(\frac{14}{(3x+4)(3x-2)}\) is negative when the denominator \((3x + 4)(3x - 2)\) is negative.

### Step 6: Find Critical Points
Set the denominator equal to zero to find critical points:
1. \(3x + 4 = 0 \Rightarrow x = -\frac{4}{3}\)
2. \(3x - 2 = 0 \Rightarrow x = \frac{2}{3}\)

### Step 7: Test Intervals
We will test the intervals determined by the critical points \(-\frac{4}{3}\) and \(\frac{2}{3}\):
1. Interval \((- \infty, -\frac{4}{3})\)
2. Interval \((- \frac{4}{3}, \frac{2}{3})\)
3. Interval \((\frac{2}{3}, \infty)\)

**Testing the intervals:**
- For \(x < -\frac{4}{3}\) (e.g., \(x = -2\)):
\((3(-2) + 4)(3(-2) - 2) = (-6 + 4)(-6 - 2) = (-2)(-8) > 0\)

- For \(-\frac{4}{3} < x < \frac{2}{3}\) (e.g., \(x = 0\)):
\((3(0) + 4)(3(0) - 2) = (4)(-2) < 0\)

- For \(x > \frac{2}{3}\) (e.g., \(x = 1\)):
\((3(1) + 4)(3(1) - 2) = (3 + 4)(3 - 2) = (7)(1) > 0\)

### Step 8: Conclusion
The expression \(\frac{14}{(3x+4)(3x-2)} < 0\) is satisfied in the interval:
\[ \left(-\frac{4}{3}, \frac{2}{3}\right) \]

### Final Answer
The values of \(x\) that satisfy the inequality are:
\[ x \in \left(-\frac{4}{3}, \frac{2}{3}\right) \]