**Introduction:**

To find the number of integral solutions for the equation |(x-3)(x-2)| = -(x-3)(x-2), we need to solve the equation and determine the values of x that satisfy the inequality.

**Solving the Equation:**

Let's break down the equation step by step:

1. |(x-3)(x-2)| = -(x-3)(x-2)

2. Since the absolute value of a quantity is always non-negative, the left side of the equation must be non-negative. Therefore, for the equation to be true, the right side must also be non-negative.

3. -(x-3)(x-2) is non-negative when the expression (x-3)(x-2) is non-positive or when it equals zero.

4. To find the integral solutions, we need to solve two separate equations:

a) (x-3)(x-2) ≤ 0
b) (x-3)(x-2) = 0

**Solving Inequality (x-3)(x-2) ≤ 0:**

To solve the inequality (x-3)(x-2) ≤ 0, we can use the concept of the sign chart or the number line method:

1. First, determine the critical points of the inequality by setting (x-3)(x-2) = 0 and solving for x.

(x-3)(x-2) = 0
x - 3 = 0 or x - 2 = 0
x = 3 or x = 2

2. Plot the critical points on a number line.

---|---2---3---|---

3. Choose a test point in each interval and evaluate the inequality.

For x < 2,="" choose="" x="1:" (1-3)(1-2)="(-2)(-1)" =="" 2="" /> 0
For 2 < x="" />< 3,="" choose="" x="2.5:" (2.5-3)(2.5-2)="(-0.5)(0.5)" =="" -0.25="" />< />
For x > 3, choose x = 4: (4-3)(4-2) = (1)(2) = 2 > 0

4. Determine the sign of the inequality in each interval.

---(-)---0---(+)---

5. Based on the sign chart, we can see that the inequality (x-3)(x-2) ≤ 0 holds true when x ≤ 2 or x ≥ 3.

**Solving Equation (x-3)(x-2) = 0:**

To solve the equation (x-3)(x-2) = 0, we set the expression equal to zero and solve for x:

(x-3)(x-2) = 0

Setting each factor equal to zero:

x - 3 = 0 or x - 2 = 0

x = 3 or x = 2

Therefore, the integral solutions for this equation are x = 3 and x = 2.

**Combining the Solutions:**

To find the values of x that satisfy both the inequality and the equation, we can consider the intersection of the solutions from both cases