The problem asks for the possible value of the expression $|-X| – X^2$ given the absolute value equation $|-2X + 9| = 3$. We need to first find the possible values of X and then substitute them into the expression.

Absolute Value Equation Solving

The equation $|-2X + 9| = 3$ implies two possibilities:

• Case 1: $-2X + 9 = 3$
• Case 2: $-2X + 9 = -3$

Case 1 Calculation

Solve for X in the first case:

$-2X + 9 = 3$

Subtract 9 from both sides:

$-2X = 3 - 9$

$-2X = -6$

Divide by -2:

$X = \frac{-6}{-2}$

$X = 3$

Case 2 Calculation

Solve for X in the second case:

$-2X + 9 = -3$

Subtract 9 from both sides:

$-2X = -3 - 9$

$-2X = -12$

Divide by -2:

$X = \frac{-12}{-2}$

$X = 6$

So, the possible values for X are 3 and 6.

Expression Evaluation

Now, evaluate the expression $|-X| – X^2$ for each possible value of X.

Evaluating for X = 3

Substitute $X = 3$ into the expression:

$|-3| - (3)^2$

$= 3 - 9$

$= -6$

Evaluating for X = 6

Substitute $X = 6$ into the expression:

$|-6| - (6)^2$

$= 6 - 36$

$= -30$

The possible values for the expression are -6 and -30. The option -30 is listed.