Solve for `x : (x + 2)/(4 + x) = 5/9`

Answer

To solve the equation \(\frac{x + 2}{4 + x} = \frac{5}{9}\), we will follow these steps:

### Step 1: Cross Multiply
We will cross multiply to eliminate the fractions. This means we will multiply the numerator of one side by the denominator of the other side.

\[ 9(x + 2) = 5(4 + x) \]

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

\[ 9x + 18 = 20 + 5x \]

### Step 3: Move all terms involving \(x\) to one side
Now, we will move all terms involving \(x\) to the left side and constant terms to the right side. We can do this by subtracting \(5x\) from both sides and subtracting \(18\) from both sides.

\[ 9x - 5x = 20 - 18 \]

### Step 4: Simplify
Now we will simplify both sides of the equation.

\[ 4x = 2 \]

### Step 5: Solve for \(x\)
Finally, we will divide both sides by \(4\) to solve for \(x\).

\[ x = \frac{2}{4} = \frac{1}{2} \]

Thus, the solution for the equation is:

\[ \boxed{\frac{1}{2}} \]

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