Solve :
`(i) |x^(2)-2x|le x` , `(ii) (x^(2)-9)(|x|-2)le0`

Answer

To solve the given inequalities step by step, we will break down each part of the question.

### (i) Solve the inequality: \(|x^2 - 2x| \leq x\)

**Step 1: Remove the absolute value.**
The expression \(|x^2 - 2x| \leq x\) can be split into two cases:
1. \(x^2 - 2x \leq x\)
2. \(-(x^2 - 2x) \leq x\) or \(x^2 - 2x \geq -x\)

**Step 2: Solve the first case.**
From \(x^2 - 2x \leq x\):
\[ x^2 - 3x \leq 0 \]
Factoring gives:
\[ x(x - 3) \leq 0 \]
The critical points are \(x = 0\) and \(x = 3\). We test intervals:
- For \(x < 0\) (e.g., \(x = -1\)): \((-1)(-4) > 0\) (not satisfied)
- For \(0 < x < 3\) (e.g., \(x = 1\)): \((1)(-2) < 0\) (satisfied)
- For \(x > 3\) (e.g., \(x = 4\)): \((4)(1) > 0\) (not satisfied)

Thus, the solution from this case is:
\[ 0 \leq x \leq 3 \]

**Step 3: Solve the second case.**
From \(x^2 - 2x \geq -x\):
\[ x^2 - x \geq 0 \]
Factoring gives:
\[ x(x - 1) \geq 0 \]
The critical points are \(x = 0\) and \(x = 1\). We test intervals:
- For \(x < 0\) (e.g., \(x = -1\)): \((-1)(-2) > 0\) (not satisfied)
- For \(0 < x < 1\) (e.g., \(x = 0.5\)): \((0.5)(-0.5) < 0\) (not satisfied)
- For \(x > 1\) (e.g., \(x = 2\)): \((2)(1) > 0\) (satisfied)

Thus, the solution from this case is:
\[ x \geq 1 \]

**Step 4: Combine the solutions.**
From the first case, we have \(0 \leq x \leq 3\) and from the second case \(x \geq 1\). The combined solution is:
\[ 1 \leq x \leq 3 \]

### (ii) Solve the inequality: \((x^2 - 9)(|x| - 2) \leq 0\)

**Step 1: Identify critical points.**
The critical points occur when \(x^2 - 9 = 0\) and \(|x| - 2 = 0\):
1. \(x^2 - 9 = 0 \Rightarrow x = 3, -3\)
2. \(|x| - 2 = 0 \Rightarrow x = 2, -2\)

Thus, the critical points are \(-3, -2, 2, 3\).

**Step 2: Test intervals.**
We will test the intervals defined by these critical points: \((-∞, -3)\), \((-3, -2)\), \((-2, 2)\), \((2, 3)\), and \((3, ∞)\).

- For \(x < -3\) (e.g., \(x = -4\)):
\[ (-4^2 - 9)(|-4| - 2) = (16 - 9)(4 - 2) = 7 \cdot 2 > 0 \quad \text{(not satisfied)} \]
- For \(-3 < x < -2\) (e.g., \(x = -2.5\)):
\[ ((-2.5)^2 - 9)(|-2.5| - 2) = (6.25 - 9)(2.5 - 2) = (-2.75)(0.5) < 0 \quad \text{(satisfied)} \]
- For \(-2 < x < 2\) (e.g., \(x = 0\)):
\[ (0^2 - 9)(|0| - 2) = (-9)(-2) > 0 \quad \text{(not satisfied)} \]
- For \(2 < x < 3\) (e.g., \(x = 2.5\)):
\[ ((2.5)^2 - 9)(|2.5| - 2) = (6.25 - 9)(2.5 - 2) = (-2.75)(0.5) < 0 \quad \text{(satisfied)} \]
- For \(x > 3\) (e.g., \(x = 4\)):
\[ (4^2 - 9)(|4| - 2) = (16 - 9)(4 - 2) = 7 \cdot 2 > 0 \quad \text{(not satisfied)} \]

**Step 3: Include critical points.**
The critical points where the expression equals zero are \(-3, -2, 2, 3\). Since the inequality is \(\leq 0\), we include these points.

**Final Solution:**
The solution to the inequality is:
\[ [-3, -2] \cup [2, 3] \]

### Summary of Solutions:
1. For \(|x^2 - 2x| \leq x\): \(x \in [1, 3]\)
2. For \((x^2 - 9)(|x| - 2) \leq 0\): \(x \in [-3, -2] \cup [2, 3]\)