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Chapter 2: Problem 2

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$(x+3)(x-5)>0$$

Short Answer

Expert verified
The solution set for the inequality \((x+3)(x-5)>0\) is \((- 鈭� ,-3)鈭�(5 ,+ 鈭�)\).

Step by step solution

01

Find the Roots

Set the polynomial part of the equation equal to zero and solve. This gives: \(x+3=0\) 鈫� \(x=-3\), and \(x-5=0\) 鈫� \(x=5\). So, -3 and 5 are roots of the original equation.
02

Test for solution intervals on the number line

Draw a number line and mark the roots -3 and 5. The roots divide the line into three intervals: \(x<-3\), \(-3 < x <5\) and \(x>5\). Choose a test point from each interval. Let's pick -4 from first interval, 0 from second and 6 from third interval, and see if they satisfy the inequality.
03

Express the result in Interval notation

Given the results of the tests, we determine the range of \(x\) where the inequality holds. The test point -4 gives a positive result, so \(x<-3\) is part of the solution. The test point 0 gives a negative result, hence the interval \(-3 < x < 5\) is not part of the solution. The test point 6 gives a positive result, so \(x > 5\) is part of the solution. Therefore, the solution of the inequality in interval notation is \((- 鈭� ,-3)鈭�(5 ,+ 鈭�)\).

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