Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x+1)(x+2)(x+3)≥0

Verified step by step guidance

1. First, identify the critical points by setting each factor equal to zero: solve \(x+1=0\), \(x+2=0\), and \(x+3=0\). These points divide the real number line into intervals.
2. The critical points are \(x = -1\), \(x = -2\), and \(x = -3\). These points split the number line into four intervals: \(( -\infty, -3 )\), \(( -3, -2 )\), \(( -2, -1 )\), and \(( -1, \infty )\).
3. Choose a test point from each interval and substitute it into the inequality \((x+1)(x+2)(x+3) \geq 0\) to determine if the product is positive or negative in that interval.
4. Based on the sign of the product in each interval, determine which intervals satisfy the inequality \(\geq 0\). Remember to include the critical points where the product equals zero because the inequality is \(\geq 0\), not just \(>0\).
5. Express the solution set as a union of intervals where the inequality holds true, and then graph these intervals on the real number line, marking included endpoints with solid dots.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols (>, <, ≥, ≤). Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Critical Points and Sign Analysis

Critical points are the values of the variable that make each factor of the polynomial zero. These points divide the number line into intervals. By testing values from each interval, you determine whether the polynomial is positive or negative there, which helps identify where the inequality holds.

Interval Notation and Graphing Solutions

Interval notation expresses the solution set as intervals on the real number line, using parentheses for excluded endpoints and brackets for included ones. Graphing the solution on a number line visually represents these intervals, showing where the polynomial inequality is satisfied.