(x-4)%2Fx%2B3-0.jpeg "(x-2)(x-4)/x+3 0")
Solving the Inequality: (x-2)(x-4)/(x+3) < 0
This problem involves solving a rational inequality. Here's a step-by-step guide:
1. Find the critical points
The critical points are the values of x that make the expression equal to zero or undefined.
- Numerator: The numerator, (x-2)(x-4), equals zero when x = 2 or x = 4.
- Denominator: The denominator, (x+3), equals zero when x = -3. This value makes the expression undefined.
Therefore, our critical points are x = -3, x = 2, and x = 4.
2. Create a sign chart
We'll use a sign chart to determine the sign of the expression in different intervals:
| Interval | x < -3 | -3 < x < 2 | 2 < x < 4 | x > 4 |
|---|---|---|---|---|
| x + 3 | - | + | + | + |
| x - 2 | - | - | + | + |
| x - 4 | - | - | - | + |
| (x-2)(x-4)/(x+3) | + | - | + | - |
Explanation:
- x + 3: The expression is negative when x < -3 and positive when x > -3.
- x - 2: The expression is negative when x < 2 and positive when x > 2.
- x - 4: The expression is negative when x < 4 and positive when x > 4.
Multiplying the signs in each interval gives us the sign of the overall expression.
3. Determine the solution
We want the expression to be less than zero (negative). Looking at our sign chart, the solution is:
-3 < x < 2 or x > 4
4. Express the solution in interval notation
The solution in interval notation is:
(-3, 2) U (4, ∞)
Important Notes
- The critical points x = -3, x = 2, and x = 4 are not included in the solution because the expression is equal to zero at these points.
- We are looking for where the expression is strictly less than zero, not less than or equal to zero.