Solve each of the following equatins :
`(5)/(2x+1)+(6)/(x+1)=3`

Answer

To solve the equation \(\frac{5}{2x+1} + \frac{6}{x+1} = 3\), we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
The LCM of the denominators \(2x + 1\) and \(x + 1\) is \((2x + 1)(x + 1)\).

### Step 2: Multiply through by the LCM
Multiply the entire equation by the LCM to eliminate the fractions:
\[ (2x + 1)(x + 1) \left(\frac{5}{2x+1} + \frac{6}{x+1}\right) = 3(2x + 1)(x + 1) \]
This simplifies to:
\[ 5(x + 1) + 6(2x + 1) = 3(2x^2 + 3x + 1) \]

### Step 3: Expand both sides
Expanding the left side:
\[ 5x + 5 + 12x + 6 = 3(2x^2 + 3x + 1) \]
Combining like terms gives:
\[ 17x + 11 = 3(2x^2 + 3x + 1) \]

Expanding the right side:
\[ 17x + 11 = 6x^2 + 9x + 3 \]

### Step 4: Rearrange the equation
Rearranging the equation gives:
\[ 0 = 6x^2 + 9x + 3 - 17x - 11 \]
This simplifies to:
\[ 0 = 6x^2 - 8x - 8 \]

### Step 5: Divide the entire equation by 2
To simplify, divide the entire equation by 2:
\[ 0 = 3x^2 - 4x - 4 \]

### Step 6: Factor the quadratic equation
We will factor the quadratic equation:
\[ 3x^2 - 6x + 2x - 4 = 0 \]
Grouping gives:
\[ 3x(x - 2) + 2(x - 2) = 0 \]
Factoring out \((x - 2)\):
\[ (x - 2)(3x + 2) = 0 \]

### Step 7: Solve for \(x\)
Setting each factor to zero gives:
1. \(x - 2 = 0 \Rightarrow x = 2\)
2. \(3x + 2 = 0 \Rightarrow x = -\frac{2}{3}\)

### Final Solution
The solutions for the equation are:
\[ x = 2 \quad \text{and} \quad x = -\frac{2}{3} \]

---