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−4)(x+2)>0

Verified step by step guidance

1. Start by identifying the critical points of the inequality \((x-4)(x+2) > 0\). These are the values of \(x\) that make each factor equal to zero, so set each factor equal to zero: \(x - 4 = 0\) and \(x + 2 = 0\).
2. Solve each equation to find the critical points: \(x = 4\) and \(x = -2\). These points divide the real number line into three intervals: \((-\infty, -2)\), \((-2, 4)\), and \((4, \infty)\).
3. Test a value from each interval in the original inequality \((x-4)(x+2) > 0\) to determine if the product is positive or negative in that interval. For example, pick \(x = -3\) for \((-\infty, -2)\), \(x = 0\) for \((-2, 4)\), and \(x = 5\) for \((4, \infty)\).
4. Determine the sign of the product in each interval by substituting the test values into \((x-4)(x+2)\). If the product is greater than zero, that interval is part of the solution set; if not, it is excluded.
5. Express the solution set as the union of intervals where the inequality holds true, and write it in interval notation. Also, graph these intervals on the real number line, using open circles at the critical points since the inequality is strict (greater than zero, not greater than or equal to).

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols like >, <, ≥, or ≤. 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 values of the variable where the polynomial equals zero, dividing the number line into intervals. By testing points in each interval, you determine whether the polynomial is positive or negative there, which helps identify where the inequality holds true.

Interval Notation and Graphing on the Number Line

Interval notation expresses solution sets as ranges of values using parentheses and brackets to indicate open or closed intervals. Graphing these solutions on a number line visually represents where the inequality is satisfied, aiding in understanding and communication of the solution.