To solve the problem of finding the number of integer values of (x) satisfying (|x - 3| + |x + 2| \leq 7), we analyze the absolute value expression by considering different cases based on the critical points (-2) and (3):

Case 1: (x \leq -2)

The expression becomes ((3 - x) + (-x - 2) = 1 - 2x).
Set (1 - 2x \leq 7):
(-2x \leq 6 \implies x \geq -3).
Valid integer values: (x = -3, -2) (2 values).

Case 2: (-2 < x < 3)

The expression simplifies to ((3 - x) + (x + 2) = 5), which is always (\leq 7).
Valid integer values: (x = -1, 0, 1, 2) (4 values).

Case 3: (x \geq 3)

The expression becomes ((x - 3) + (x + 2) = 2x - 1).
Set (2x - 1 \leq 7):
(2x \leq 8 \implies x \leq 4).
Valid integer values: (x = 3, 4) (2 values).

Total number of integer values: (2 + 4 + 2 = 8).

Answer: (\boxed{8})

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