To solve this inequality, we can start by expanding the square term:

(3y-7)² = (3y-7)(3y-7) = 9y² - 42y + 49

Now we can rewrite the inequality:

9y² - 42y + 49 - y(y-1) < 0

Simplifying further:

9y² - 42y + 49 - y² + y < 0

Combining like terms:

8y² - 41y + 49 < 0

Now we can solve this quadratic inequality. We can start by finding the roots of the quadratic equation:

8y² - 41y + 49 = 0

Using the quadratic formula:

y = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 8, b = -41, and c = 49. Plugging in these values:

y = (-(-41) ± √((-41)² - 4(8)(49))) / (2(8))

Simplifying:

y = (41 ± √(1681 - 1568)) / 16

y = (41 ± √113) / 16

So the roots of the quadratic equation are:

y₁ = (41 + √113) / 16 ≈ 2.53
y₂ = (41 - √113) / 16 ≈ 0.31

Now we can plot these roots on a number line:

-------------------0.31------------------2.53-------------------

Next, we need to determine the sign of the quadratic expression in each interval. We can choose a test point in each interval to determine the sign.

For example, let's choose y = 0 as a test point in the interval (-∞, 0.31). Plugging in y = 0 into the quadratic expression:

8(0)² - 41(0) + 49 < 0

49 < 0

Since this is false, the quadratic expression is negative in the interval (-∞, 0.31).

Now let's choose y = 1 as a test point in the interval (0.31, 2.53). Plugging in y = 1 into the quadratic expression:

8(1)² - 41(1) + 49 < 0

16 - 41 + 49 < 0

24 < 0

Since this is false, the quadratic expression is also negative in the interval (0.31, 2.53).

Finally, let's choose y = 3 as a test point in the interval (2.53, ∞). Plugging in y = 3 into the quadratic expression:

8(3)² - 41(3) + 49 < 0

72 - 123 + 49 < 0

-2 < 0

Since this is true, the quadratic expression is positive in the interval (2.53, ∞).

Now we can determine the solution to the inequality by looking at the sign of the quadratic expression in each interval:

(-∞, 0.31): Negative
(0.31, 2.53): Negative
(2.53, ∞): Positive

Since the inequality is less than 0, we are looking for the intervals where the quadratic expression is negative. Therefore, the solution to the inequality is:

y < 0.31 or 0.31 < y < 2.53