Solve the following equation for $x$: \[ \ \frac{2}{3} = \frac{4}{x-5}.\]
To solve for $x$, I need to isolate it on one side of the equation. I can start by cross-multiplying the fractions, which means multiplying the numerator of one fraction by the denominator of the other and setting them equal. So, I get \[ 2(x-5) = 3 \cdot 4.\] Next, I can simplify the expression by distributing the 2 and the 3, and I get \[ 2x - 10 = 12.\] Now, I can add 10 to both sides to eliminate the constant term on the left, and I get \[ 2x = 22.\] Finally, I can divide both sides by 2 to get the value of $x$, and I get \[ x = 11.\] 