(x-3)(x%2B2)-0.jpeg "(x-1)(x-3)(x+2) 0")
Solving the Inequality: (x-1)(x-3)(x+2) > 0
This problem involves finding the values of x that satisfy the inequality (x-1)(x-3)(x+2) > 0. To solve this, we'll use the following steps:
1. Find the critical points
The critical points are the values of x where the expression equals zero. We set each factor to zero and solve:
- x - 1 = 0 => x = 1
- x - 3 = 0 => x = 3
- x + 2 = 0 => x = -2
These critical points divide the number line into four intervals:
- Interval 1: x < -2
- Interval 2: -2 < x < 1
- Interval 3: 1 < x < 3
- Interval 4: x > 3
2. Test each interval
We'll choose a test value within each interval and substitute it into the original inequality to determine if it satisfies the condition.
Interval 1 (x < -2): Let's try x = -3. (-3 - 1)(-3 - 3)(-3 + 2) = (-4)(-6)(-1) = -24. Since -24 is not greater than 0, this interval does not satisfy the inequality.
Interval 2 (-2 < x < 1): Let's try x = 0. (0 - 1)(0 - 3)(0 + 2) = (-1)(-3)(2) = 6. Since 6 is greater than 0, this interval satisfies the inequality.
Interval 3 (1 < x < 3): Let's try x = 2. (2 - 1)(2 - 3)(2 + 2) = (1)(-1)(4) = -4. Since -4 is not greater than 0, this interval does not satisfy the inequality.
Interval 4 (x > 3): Let's try x = 4. (4 - 1)(4 - 3)(4 + 2) = (3)(1)(6) = 18. Since 18 is greater than 0, this interval satisfies the inequality.
3. Write the solution
The inequality (x-1)(x-3)(x+2) > 0 is satisfied when:
- -2 < x < 1 or x > 3
This can also be expressed in interval notation:
(-2, 1) U (3, ∞)