Solve the Rational Equation: (5+3(x-2))/(5+x) = 3/4

To solve this problem, we'll follow these steps:

• Step 1: Cross-multiply to eliminate the fractions.
• Step 2: Simplify the resulting linear equation.
• Step 3: Solve for $x$.

Now, let's work through each step:

Step 1: Starting with the equation:

$\frac{5 + 3 \times (x - 2)}{5 + x} = \frac{3}{4}$

Cross-multiply to remove the fractions:

$4 \cdot (5 + 3 \times (x - 2)) = 3 \cdot (5 + x)$

Step 2: Simplify the equation. Expand inside the brackets:

$4 \cdot (5 + 3x - 6) = 3 \cdot (5 + x)$

Simplify further:

$4 \cdot (3x - 1) = 3 \cdot (5 + x)$

Distribute the constants:

$12x - 4 = 15 + 3x$

Step 3: Solve for $x$.

Subtract $3x$ from both sides:

$12x - 3x - 4 = 15$

$9x - 4 = 15$

Add 4 to both sides:

$9x = 19$

Divide both sides by 9:

$x = \frac{19}{9}$

Therefore, the solution to the problem is $x = \frac{19}{9}$.