To solve the inequality \((|x-1|-3)(|x+2|-5) < 0\), we need to analyze the expression step by step.
### Step 1: Identify the critical points
The critical points occur when each absolute value expression equals the constants. We set up the equations:
1. \(|x-1| - 3 = 0\) leads to \(|x-1| = 3\)
- This gives us two cases:
- \(x - 1 = 3 \implies x = 4\)
- \(x - 1 = -3 \implies x = -2\)
2. \(|x+2| - 5 = 0\) leads to \(|x+2| = 5\)
- This gives us two cases:
- \(x + 2 = 5 \implies x = 3\)
- \(x + 2 = -5 \implies x = -7\)
The critical points are \(x = -7, -2, 3, 4\).
### Step 2: Test intervals between critical points
We will test the sign of the expression \((|x-1|-3)(|x+2|-5)\) in the intervals defined by the critical points:
1. \((-∞, -7)\)
2. \((-7, -2)\)
3. \((-2, 3)\)
4. \((3, 4)\)
5. \((4, ∞)\)
### Step 3: Evaluate each interval
1. **Interval \((-∞, -7)\)**: Choose \(x = -8\)
- \(|-8-1| - 3 = 9 - 3 = 6\) (positive)
- \(|-8+2| - 5 = 6 - 5 = 1\) (positive)
- Product: \(6 \cdot 1 > 0\)
2. **Interval \((-7, -2)\)**: Choose \(x = -5\)
- \(|-5-1| - 3 = 6 - 3 = 3\) (positive)
- \(|-5+2| - 5 = 3 - 5 = -2\) (negative)
- Product: \(3 \cdot (-2) < 0\)
3. **Interval \((-2, 3)\)**: Choose \(x = 0\)
- \(|0-1| - 3 = 1 - 3 = -2\) (negative)
- \(|0+2| - 5 = 2 - 5 = -3\) (negative)
- Product: \((-2) \cdot (-3) > 0\)
4. **Interval \((3, 4)\)**: Choose \(x = 3.5\)
- \(|3.5-1| - 3 = 2.5 - 3 = -0.5\) (negative)
- \(|3.5+2| - 5 = 5.5 - 5 = 0.5\) (positive)
- Product: \((-0.5) \cdot 0.5 < 0\)
5. **Interval \((4, ∞)\)**: Choose \(x = 5\)
- \(|5-1| - 3 = 4 - 3 = 1\) (positive)
- \(|5+2| - 5 = 7 - 5 = 2\) (positive)
- Product: \(1 \cdot 2 > 0\)
### Step 4: Combine results
The expression is negative in the intervals:
- \((-7, -2)\)
- \((3, 4)\)
### Step 5: Write the final solution
Thus, the solution to the inequality \((|x-1|-3)(|x+2|-5) < 0\) is:
\[
x \in (-7, -2) \cup (3, 4)
\]
