Solve : `(t + 2)/(3) + 1/(t + 1) = (t - 3)/(2) - (t - 1)/(6)`

Options

A. `t=1 `
B. `t=-2`
C. `t= 2 `
D. `t= 0 `

Answer

To solve the equation

\[ \frac{t + 2}{3} + \frac{1}{t + 1} = \frac{t - 3}{2} - \frac{t - 1}{6} \]

we will follow these steps:

### Step 1: Find the LCM of the denominators
The denominators on the left side are \(3\) and \(t + 1\), and on the right side are \(2\) and \(6\).

The LCM of \(3\) and \(t + 1\) is \(3(t + 1)\) and the LCM of \(2\) and \(6\) is \(6\).

### Step 2: Rewrite the equation using the LCM
Multiply both sides of the equation by \(6(t + 1)\) to eliminate the fractions:

\[ 6(t + 1) \left(\frac{t + 2}{3}\right) + 6(t + 1) \left(\frac{1}{t + 1}\right) = 6(t + 1) \left(\frac{t - 3}{2}\right) - 6(t + 1) \left(\frac{t - 1}{6}\right) \]

This simplifies to:

\[ 2(t + 1)(t + 2) + 6 = 3(t + 1)(t - 3) - (t + 1)(t - 1) \]

### Step 3: Expand both sides
Now we expand both sides:

Left side:
\[ 2(t^2 + 3t + 2) + 6 = 2t^2 + 6t + 4 + 6 = 2t^2 + 6t + 10 \]

Right side:
\[ 3(t^2 - 2t - 3) - (t^2 - 0) = 3t^2 - 6t - 9 - t^2 + 1 = 2t^2 - 6t - 8 \]

### Step 4: Set the equation to zero
Now we set both sides equal to each other:

\[ 2t^2 + 6t + 10 = 2t^2 - 6t - 8 \]

Subtract \(2t^2\) from both sides:

\[ 6t + 10 = -6t - 8 \]

### Step 5: Combine like terms
Add \(6t\) to both sides:

\[ 12t + 10 = -8 \]

Now, subtract \(10\) from both sides:

\[ 12t = -18 \]

### Step 6: Solve for \(t\)
Divide both sides by \(12\):

\[ t = -\frac{18}{12} = -\frac{3}{2} \]

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

\[ t = -\frac{3}{2} \]

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