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Chapter 2: Problem 1

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$(x-4)(x+2)>0$$

Short Answer

Expert verified
The solution is \((-\infty, -2) \cup (4, \infty)\).

Step by step solution

01

Find the Zero Points

Set the polynomial equal to zero to find the critical points. We find the zero points by setting each factor equal to zero. For this we have \(x - 4 = 0\) and \(x + 2 = 0\). Solving these we get \(x = 4\) and \(x = -2\) these are the points that divide the number line.
02

Test the Intervals

Now, we test the sign of the function in each interval. The critical points divide the real line into three intervals: \((-\infty, -2)\), \((-2, 4)\), and \( (4, \infty)\). Select a test point from each interval, say -3, 0, and 5. If we substitute these points in our inequality (x-4)(x+2)>0, at x=-3 and x=5 the product will be greater than zero (>0), which means in those intervals the inequality holds. At x=0, the product will be less than zero (<0), hence the inequality does not hold in that interval.
03

Write the interval notation

The solution set includes all the x values in the intervals for which the inequality holds. The inequality doesn't hold at x = -2 and x = 4, so parentheses should be used. This leads to the solution in interval notation as \((-\infty, -2) \cup (4, \infty)\).

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