Find the roots of the equation : `(1)/(x+4)-(1)/(x-7)=(11)/(30)(xne-4,7)`

Answer

To find the roots of the equation

\[ \frac{1}{x+4} - \frac{1}{x-7} = \frac{11}{30} \]

we will follow these steps:

### Step 1: Find a common denominator
To combine the left side of the equation, we need to find a common denominator, which is \((x + 4)(x - 7)\).

\[ \frac{(x - 7) - (x + 4)}{(x + 4)(x - 7)} = \frac{11}{30} \]

### Step 2: Simplify the numerator
Now, simplify the numerator:

\[ (x - 7) - (x + 4) = x - 7 - x - 4 = -11 \]

So, we can rewrite the equation as:

\[ \frac{-11}{(x + 4)(x - 7)} = \frac{11}{30} \]

### Step 3: Cross-multiply
Now we will cross-multiply to eliminate the fractions:

\[ -11 \cdot 30 = 11 \cdot (x + 4)(x - 7) \]

This simplifies to:

\[ -330 = 11(x + 4)(x - 7) \]

### Step 4: Divide both sides by 11
Next, divide both sides by 11:

\[ -30 = (x + 4)(x - 7) \]

### Step 5: Expand the right side
Now, expand the right side:

\[ -30 = x^2 - 7x + 4x - 28 \]

This simplifies to:

\[ -30 = x^2 - 3x - 28 \]

### Step 6: Move all terms to one side
Now, move all terms to one side of the equation:

\[ x^2 - 3x - 28 + 30 = 0 \]

This simplifies to:

\[ x^2 - 3x + 2 = 0 \]

### Step 7: Factor the quadratic equation
Now, we will factor the quadratic equation:

\[ x^2 - 3x + 2 = (x - 1)(x - 2) = 0 \]

### Step 8: Solve for x
Setting each factor to zero gives us:

\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

### Final Answer
Thus, the roots of the equation are:

\[ x = 1 \quad \text{and} \quad x = 2 \]

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