To solve this equation, we can start by taking the natural logarithm of both sides to eliminate the exponents:

ln(1.5^(5x-7)) = ln((2/3)^(x+1))

Using the property of logarithms that ln(a^b) = b*ln(a), we can simplify this to:

(5x-7)ln(1.5) = (x+1)ln(2/3)

Now, we can distribute the logarithms on both sides:

5xln(1.5) - 7ln(1.5) = xln(2/3) + ln(2/3)

Next, we can isolate the x terms by moving everything not containing x to the other side of the equation:

5xln(1.5) - xln(2/3) = 7ln(1.5) + ln(2/3)

Factor out x on the left side:

x(5ln(1.5) - ln(2/3)) = 7ln(1.5) + ln(2/3)

Divide both sides by (5ln(1.5) - ln(2/3)) to solve for x:

x = (7ln(1.5) + ln(2/3))/(5ln(1.5) - ln(2/3))

By simplifying the expression further, we can find the numerical value of x:

x ≈ 2.117