`(x+5)/(4)=(1-x)/(3x-4)`
If the solutions to the equation above are r and s with `rgts`, what is the value of r-s?

Options

A. `sqrt(7)`
B. `(5)/(2)`
C. `sqrt(57)`
D. `5`

Answer

To solve the equation \(\frac{x+5}{4} = \frac{1-x}{3x-4}\) and find the value of \(r - s\) where \(r\) and \(s\) are the solutions with \(r > s\), we will follow these steps:

### Step 1: Cross Multiply
We start by cross-multiplying to eliminate the fractions:
\[ (x + 5)(3x - 4) = (1 - x)(4) \]

### Step 2: Expand Both Sides
Now, we expand both sides of the equation:
\[ 3x^2 - 4x + 15x - 20 = 4 - 4x \]
This simplifies to:
\[ 3x^2 + 11x - 20 = 4 - 4x \]

### Step 3: Rearrange the Equation
Next, we rearrange the equation to bring all terms to one side:
\[ 3x^2 + 11x + 4x - 20 - 4 = 0 \]
This simplifies to:
\[ 3x^2 + 15x - 24 = 0 \]

### Step 4: Identify Coefficients
We identify the coefficients \(a\), \(b\), and \(c\) from the standard form \(ax^2 + bx + c = 0\):
- \(a = 3\)
- \(b = 15\)
- \(c = -24\)

### Step 5: Apply the Quadratic Formula
We apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ x = \frac{-15 \pm \sqrt{15^2 - 4 \cdot 3 \cdot (-24)}}{2 \cdot 3} \]

### Step 6: Calculate the Discriminant
Calculating the discriminant:
\[ 15^2 = 225 \]
\[ -4 \cdot 3 \cdot (-24) = 288 \]
Thus,
\[ b^2 - 4ac = 225 + 288 = 513 \]

### Step 7: Substitute Back into the Formula
Now substituting back into the formula:
\[ x = \frac{-15 \pm \sqrt{513}}{6} \]

### Step 8: Simplify the Roots
The roots can be expressed as:
\[ x_1 = \frac{-15 + \sqrt{513}}{6}, \quad x_2 = \frac{-15 - \sqrt{513}}{6} \]
Let \(r = x_1\) and \(s = x_2\) where \(r > s\).

### Step 9: Calculate \(r - s\)
Now we find \(r - s\):
\[ r - s = \left(\frac{-15 + \sqrt{513}}{6}\right) - \left(\frac{-15 - \sqrt{513}}{6}\right) \]
This simplifies to:
\[ r - s = \frac{-15 + \sqrt{513} + 15 + \sqrt{513}}{6} = \frac{2\sqrt{513}}{6} = \frac{\sqrt{513}}{3} \]

### Step 10: Final Simplification
Since \(\sqrt{513} = 3\sqrt{57}\), we have:
\[ r - s = \frac{3\sqrt{57}}{3} = \sqrt{57} \]

### Final Answer
Thus, the value of \(r - s\) is:
\[ \sqrt{57} \]