Problem 1 - Algebra and Trigonometry

Question

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. $$ (x-4)(x+2)>0 $$

Step-by-Step Solution

Answer

The solution to the inequality is represented by the interval \(-\infty, -2) \cup (4, \infty\).

Step 1: Identify the x-values where the polynomial equals zero

Set each factor of the polynomial equal to zero and solve for x: \(x - 4 = 0\) and \(x + 2 = 0\), which gives \(x = 4\) and \(x = -2\). These are the critical points that divide the number line into intervals.

Step 2: Choose test points from each interval and determine if they satisfy the inequality

The critical points split the real number line into three intervals: \(-\infty, -2\), \(-2, 4\), and \(4, \infty\). Choose a test point from each interval, for instance, -3 for \(-\infty, -2\), 0 for \(-2, 4\), and 5 for \(4, \infty\). Plug these test points into the inequality: \((-3 -4)(-3 +2)\), \((0-4)(0+2)\), and \((5-4)(5+2)\). Only the results of the first and third tests are greater than zero which means the intervals \(-\infty, -2\) and \(4, \infty\) satisfy the inequality.

Step 3: Draw the solutions on a number line and express the solution set in interval notation

Plot the points x = -2 and x = 4 on the number line. Because the inequality is greater than zero, the points -2 and 4 are not included in the solution. Therefore, a parenthesis should be used at points -2 and 4 when expressing the solution set in interval notation. This gives the solution as \(-\infty, -2) \cup (4, \infty\).