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Chapter 3: Problem 2

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. $$ (x+3)(x-5)>0 $$

Short Answer

Expert verified
The solution to the inequality \( (x+3)(x-5)>0 \) is \( (-\infty,-3) \cup (5,\infty) \).

Step by step solution

01

Solve the inequality

First, we set the polynomial \( (x+3)(x-5) = 0 \) to find the points where the polynomial equals zero. The solutions are \( x = -3 \) and \( x = 5 \). These are the critical points that divide the real line into three intervals, \(-\infty,-3\), \(-3,5\) and \(5,\infty\).
02

Test critical points

Next, we choose a test point from each interval and substitute it into the inequality to see if it's satisfied. For interval \( -\infty,-3 \), select \( x= -4 \), then the inequality becomes \((-4 + 3)(-4 -5) > 0 \) which simplifies to \(-1*-9 > 0\), and this is true so all x-values in the interval \( -\infty,-3 \) are solutions. For interval \( -3,5 \), select \( x= 0 \), then the inequality becomes \(3*-5>0\), which is false, so this interval will not be included in the solution set. For interval \( 5, \infty \), select \( x= 6 \), then the inequality becomes \(9*1>0\), which is true so all x-values in the interval \(5,\infty\) are solutions.
03

Write the answer in interval notation

The solution to the inequality in interval notation is \( (-\infty,-3) \cup (5,\infty) \), meaning it's either less than -3 or greater than 5. Exclude -3 and 5 because the inequality is 'greater than' and not 'greater than or equal to'.
04

Graph the solution

To graph the solution on a number line, represent the intervals \(-\infty, -3\) and \(5, \infty\) in line graph with open circles at points -3 and 5, indicating that these points aren't included in the solution.

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