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+3)(x−5)>0

Verified step by step guidance

1. Start by identifying the critical points of the inequality \((x+3)(x-5) > 0\). These are the values of \(x\) that make each factor equal to zero, so solve \(x+3=0\) and \(x-5=0\) to find the critical points.
2. The critical points divide the real number line into intervals. In this case, the intervals are \((-\infty, -3)\), \((-3, 5)\), and \((5, \infty)\). We will test each interval to determine where the product \((x+3)(x-5)\) is positive.
3. Choose a test point from each interval and substitute it into the expression \((x+3)(x-5)\). For example, pick \(x = -4\) for \((-\infty, -3)\), \(x = 0\) for \((-3, 5)\), and \(x = 6\) for \((5, \infty)\). Determine the sign of the product at each test point.
4. Based on the signs from the test points, identify which intervals satisfy the inequality \((x+3)(x-5) > 0\). Remember, the inequality is strict, so the critical points themselves are not included in the solution set.
5. Express the solution set as a union of intervals where the product is positive, and then graph this solution on a real number line, marking open circles at the critical points \(x = -3\) and \(x = 5\) to indicate they are not included.

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 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 using parentheses and brackets to indicate open or closed intervals. Graphing on the number line visually represents these intervals, showing where the inequality is satisfied, with open or closed dots indicating whether endpoints are included.