Understanding the Absolute Value Equation

The problem requires us to first solve the absolute value equation |9y - 6| = 3. An absolute value equation of the form |x| = a (where a is a positive number) means that x is equal to either a or -a.

Applying this to our equation, we have two possible cases:

Solving for 'y'

Case 1: The expression inside the absolute value is positive or zero.

$$9y - 6 = 3$$

To solve for y, we first add 6 to both sides:

$$9y = 3 + 6$$

$$9y = 9$$

Now, we divide both sides by 9:

$$y = \frac{9}{9}$$

$$y = 1$$

Case 2: The expression inside the absolute value is negative.

$$9y - 6 = -3$$

Add 6 to both sides:

$$9y = -3 + 6$$

$$9y = 3$$

Divide both sides by 9:

$$y = \frac{3}{9}$$

$$y = \frac{1}{3}$$

So, we have found two possible values for y: 1 and 1/3.

Evaluating the Expression

Next, we need to find the value of the expression y2 - 4y/3 using the values of y we found.

When y = 1:

Substitute y = 1 into the expression:

$$ (1)^2 - \frac{4(1)}{3} $$

$$ 1 - \frac{4}{3} $$

To subtract, find a common denominator:

$$ \frac{3}{3} - \frac{4}{3} $$

$$ -\frac{1}{3} $$

When y = 1/3:

Substitute y = 1/3 into the expression:

$$ \left(\frac{1}{3}\right)^2 - \frac{4\left(\frac{1}{3}\right)}{3} $$

First, calculate the square:

$$ \frac{1}{9} - \frac{\frac{4}{3}}{3} $$

Simplify the second term:

$$ \frac{1}{9} - \frac{4}{3 \times 3} $$

$$ \frac{1}{9} - \frac{4}{9} $$

$$ -\frac{3}{9} $$

Simplify the fraction:

$$ -\frac{1}{3} $$

Conclusion

In both cases, whether y = 1 or y = 1/3, the value of the expression y2 - 4y/3 is -1/3.