(2x+1)+3/2x+1=4

3 min read Jun 16, 2024

Solving the Equation: (2x + 1) + 3 / (2x + 1) = 4

This equation presents a challenge because of the variable in the denominator. To solve it, we'll employ a combination of algebraic techniques:

1. Combining Terms

First, we need to simplify the left side of the equation. Notice that the expression can be rewritten as:

(2x + 1) + (3 / (2x + 1)) = 4

This highlights that the second term is a fraction.

2. Finding a Common Denominator

To combine the terms, we need a common denominator. We can achieve this by multiplying the first term by (2x + 1) / (2x + 1):

(2x + 1) * (2x + 1) / (2x + 1) + (3 / (2x + 1)) = 4

This simplifies to:

(4x² + 4x + 1) / (2x + 1) + (3 / (2x + 1)) = 4

Now, we can combine the numerators:

(4x² + 4x + 4) / (2x + 1) = 4

3. Multiplying Both Sides

To eliminate the fraction, we multiply both sides of the equation by (2x + 1):

(4x² + 4x + 4) = 4(2x + 1)

4. Expanding and Rearranging

Now, we expand the right side and move all terms to the left:

4x² + 4x + 4 = 8x + 4

4x² - 4x = 0

5. Factoring

We can factor out a 4x from the left side:

4x(x - 1) = 0

6. Solving for x

For the product of two terms to equal zero, at least one of them must be zero. This leads us to two potential solutions:

4x = 0 => x = 0
x - 1 = 0 => x = 1

7. Checking for Extraneous Solutions

It's crucial to check if these solutions are valid by plugging them back into the original equation. We need to make sure that the denominator (2x + 1) does not become zero:

For x = 0, the denominator is 2(0) + 1 = 1, which is valid.
For x = 1, the denominator is 2(1) + 1 = 3, which is also valid.

Therefore, both solutions x = 0 and x = 1 are valid solutions to the equation (2x + 1) + 3 / (2x + 1) = 4.