Solve each inequality. Give the solution set in interval notation. (x-4)(x-1)(x+2)>0

Verified step by step guidance

1. Identify the critical points by setting each factor equal to zero: solve \(x - 4 = 0\), \(x - 1 = 0\), and \(x + 2 = 0\). These points divide the number line into intervals.
2. The critical points are \(x = 4\), \(x = 1\), and \(x = -2\). Use these points to split the real number line into four intervals: \(( -\infty, -2 )\), \((-2, 1)\), \((1, 4)\), and \((4, \infty)\).
3. Choose a test point from each interval and substitute it into the inequality \((x-4)(x-1)(x+2) > 0\) to determine if the product is positive or negative in that interval.
4. Based on the sign of the product in each interval, select the intervals where the inequality \((x-4)(x-1)(x+2) > 0\) holds true (i.e., where the product is positive).
5. Express the solution set as a union of the intervals where the inequality is satisfied, using interval notation.

Key Concepts

Factoring and Zero-Product Property

Factoring breaks down an expression into simpler factors, making it easier to analyze. The zero-product property states that if a product of factors equals zero, at least one factor must be zero. Here, the inequality is already factored, so identifying the zeros of each factor helps find critical points.

Sign Analysis on Intervals

Sign analysis involves determining whether the product of factors is positive or negative on intervals defined by the zeros. By testing points in each interval, you can decide where the inequality holds true. This method is essential for solving polynomial inequalities.

Interval Notation

Interval notation is a concise way to represent sets of numbers between two endpoints. It uses parentheses for open intervals (excluding endpoints) and brackets for closed intervals (including endpoints). Expressing solutions in interval notation clearly communicates the solution set.