The range of values of x for which the inequality `(x-1)/(4x+5) lt (x-3)/(4x-3)` holds is

Options

A. `((-4)/(3). (5)/(8))`
B. `((-4)/(3), (1)/(2))`
C. `((-5)/(4), (3)/(4))`
D. (3, 8)

Answer

To solve the inequality \(\frac{x-1}{4x+5} < \frac{x-3}{4x-3}\), we will follow these steps:

### Step 1: Rearrange the Inequality
We start by bringing all terms to one side of the inequality:

\[ \frac{x-1}{4x+5} - \frac{x-3}{4x-3} < 0 \]

### Step 2: Find a Common Denominator
The common denominator for the fractions is \((4x + 5)(4x - 3)\). We rewrite the inequality:

\[ \frac{(x-1)(4x-3) - (x-3)(4x+5)}{(4x+5)(4x-3)} < 0 \]

### Step 3: Expand the Numerator
Now we expand the numerator:

1. For \((x-1)(4x-3)\):
\[ 4x^2 - 3x - 4x + 3 = 4x^2 - 7x + 3 \]

2. For \((x-3)(4x+5)\):
\[ 4x^2 + 5x - 12x - 15 = 4x^2 - 7x - 15 \]

Now substitute back into the inequality:

\[ \frac{(4x^2 - 7x + 3) - (4x^2 - 7x - 15)}{(4x+5)(4x-3)} < 0 \]

### Step 4: Simplify the Numerator
The \(4x^2\) terms cancel out:

\[ \frac{(4x^2 - 7x + 3) - (4x^2 - 7x - 15)}{(4x+5)(4x-3)} < 0 \]
This simplifies to:

\[ \frac{3 + 15}{(4x+5)(4x-3)} < 0 \]

Thus, we have:

\[ \frac{18}{(4x+5)(4x-3)} < 0 \]

### Step 5: Analyze the Sign of the Expression
The fraction \(\frac{18}{(4x+5)(4x-3)}\) is less than zero when the denominator is negative. Therefore, we need to find when:

\[ (4x+5)(4x-3) < 0 \]

### Step 6: Find Critical Points
Set each factor to zero to find critical points:

1. \(4x + 5 = 0 \Rightarrow x = -\frac{5}{4}\)
2. \(4x - 3 = 0 \Rightarrow x = \frac{3}{4}\)

### Step 7: Test Intervals
We test the intervals determined by the critical points \(-\frac{5}{4}\) and \(\frac{3}{4}\):

1. **Interval 1**: \(x < -\frac{5}{4}\) (choose \(x = -2\)):
\[ (4(-2)+5)(4(-2)-3) = (-8+5)(-8-3) = (-3)(-11) > 0 \]

2. **Interval 2**: \(-\frac{5}{4} < x < \frac{3}{4}\) (choose \(x = 0\)):
\[ (4(0)+5)(4(0)-3) = (5)(-3) < 0 \]

3. **Interval 3**: \(x > \frac{3}{4}\) (choose \(x = 1\)):
\[ (4(1)+5)(4(1)-3) = (9)(1) > 0 \]

### Step 8: Conclusion
The inequality holds in the interval:

\[ -\frac{5}{4} < x < \frac{3}{4} \]

Thus, the solution is:

\[ \boxed{\left(-\frac{5}{4}, \frac{3}{4}\right)} \]