Solve the following equations :
`(x-1)(x+6)-(x-2)(x-3)=3`

Options

A. `1(1)/(11)`
B. `1(1)/(9)`
C. `1(1)/(7)`
D. `1(1)/(2)`

Answer

To solve the equation \((x-1)(x+6)-(x-2)(x-3)=3\), we will follow these steps:

### Step 1: Expand both products
We start by expanding the left side of the equation.

\[ (x-1)(x+6) = x^2 + 6x - x - 6 = x^2 + 5x - 6 \]

\[ (x-2)(x-3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6 \]

Now, substituting these expansions back into the equation gives us:

\[ (x^2 + 5x - 6) - (x^2 - 5x + 6) = 3 \]

### Step 2: Simplify the equation
Now we simplify the left-hand side:

\[ x^2 + 5x - 6 - x^2 + 5x - 6 = 3 \]

The \(x^2\) terms cancel out:

\[ 5x + 5x - 6 - 6 = 3 \]

This simplifies to:

\[ 10x - 12 = 3 \]

### Step 3: Isolate the variable
Next, we will isolate \(x\) by adding 12 to both sides:

\[ 10x - 12 + 12 = 3 + 12 \]

This simplifies to:

\[ 10x = 15 \]

### Step 4: Solve for \(x\)
Now, divide both sides by 10:

\[ x = \frac{15}{10} = \frac{3}{2} \]

### Step 5: Convert to mixed fraction (if needed)
The fraction \(\frac{3}{2}\) can also be expressed as a mixed number:

\[ x = 1 \frac{1}{2} \]

### Final Answer
Thus, the solution to the equation is:

\[ x = \frac{3}{2} \quad \text{or} \quad x = 1 \frac{1}{2} \]

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