Solve the equations and check your answer in the case:
`(2x+3)/(3+x)=(3)/(2)`

Options

A. x = 2
B. x = 1
C. x = 4
D. x = 3

Answer

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

### Step 1: Cross Multiply
We start by cross-multiplying the fractions to eliminate the denominators. This means we multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the numerator of the second fraction and the denominator of the first fraction.

\[ 2 \cdot (2x + 3) = 3 \cdot (3 + x) \]

### Step 2: Distribute
Next, we distribute the numbers on both sides of the equation.

\[ 4x + 6 = 9 + 3x \]

### Step 3: Rearrange the Equation
Now, we will rearrange the equation to isolate \(x\). We can do this by moving all terms involving \(x\) to one side and constant terms to the other side.

Subtract \(3x\) from both sides:

\[ 4x - 3x + 6 = 9 \]

This simplifies to:

\[ x + 6 = 9 \]

### Step 4: Solve for \(x\)
Now, we will isolate \(x\) by subtracting 6 from both sides.

\[ x = 9 - 6 \]

This gives us:

\[ x = 3 \]

### Step 5: Check the Solution
To verify our solution, we will substitute \(x = 3\) back into the original equation.

Substituting \(x = 3\):

\[ \frac{2(3) + 3}{3 + 3} = \frac{3}{2} \]

Calculating the left-hand side:

\[ \frac{6 + 3}{6} = \frac{9}{6} = \frac{3}{2} \]

Since both sides are equal, our solution is confirmed.

### Final Answer
The solution to the equation is:

\[ x = 3 \]

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