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−7)≤0

Verified step by step guidance

1. Start by identifying the critical points of the inequality by setting each factor equal to zero: solve \(x + 1 = 0\) and \(x - 7 = 0\) to find the values of \(x\) where the expression changes sign.
2. The critical points divide the real number line into intervals. These intervals are \((-\infty, -1)\), \([-1, 7]\), and \((7, \infty)\). We will test each interval to determine where the inequality \((x+1)(x-7) \leq 0\) holds true.
3. Choose a test point from each interval and substitute it into the expression \((x+1)(x-7)\). Check whether the product is less than or equal to zero for that interval.
4. Based on the sign of the product in each interval, determine which intervals satisfy the inequality. Remember to include the points where the product equals zero because the inequality is 'less than or equal to zero'.
5. Express the solution set using interval notation, combining all intervals where the inequality holds, and then graph this solution set on the real number line by shading the appropriate regions and marking the critical points.

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 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.

Interval Notation and Graphing on the Number Line

Interval notation expresses solution sets as ranges of values, using parentheses for strict inequalities and brackets for inclusive ones. Graphing on the number line visually represents these intervals, showing where the solution lies and whether endpoints are included.