(x-1)(x-2)(x-3) 0

3 min read Jun 17, 2024

Solving the Inequality: (x-1)(x-2)(x-3) < 0

This article will explore how to solve the inequality (x-1)(x-2)(x-3) < 0. This involves understanding the concept of sign analysis and applying it to a polynomial inequality.

1. Finding the Critical Points

The critical points are the values of x where the expression (x-1)(x-2)(x-3) equals zero.

Setting each factor to zero:

x - 1 = 0 => x = 1
x - 2 = 0 => x = 2
x - 3 = 0 => x = 3

These critical points divide the number line into four intervals:

x < 1
1 < x < 2
2 < x < 3
x > 3

2. Sign Analysis

We will now examine the sign of the expression (x-1)(x-2)(x-3) within each interval:

Interval 1: x < 1

(x-1) is negative
(x-2) is negative
(x-3) is negative

The product of three negative numbers is negative.

Interval 2: 1 < x < 2

(x-1) is positive
(x-2) is negative
(x-3) is negative

The product of two negative numbers and a positive number is positive.

Interval 3: 2 < x < 3

(x-1) is positive
(x-2) is positive
(x-3) is negative

The product of two positive numbers and a negative number is negative.

Interval 4: x > 3

(x-1) is positive
(x-2) is positive
(x-3) is positive

The product of three positive numbers is positive.

3. Solution

The inequality (x-1)(x-2)(x-3) < 0 is satisfied when the expression is negative. This occurs in the intervals:

x < 1
2 < x < 3

Therefore, the solution to the inequality is: x ∈ (-∞, 1) ∪ (2, 3).

Conclusion

By identifying the critical points and analyzing the sign of the expression in each interval, we can solve the inequality (x-1)(x-2)(x-3) < 0. The solution includes all values of x less than 1 and between 2 and 3.