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Chapter 2: Problem 45

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$\frac{x+3}{x+4}<0$$

Short Answer

Expert verified
The solution to the inequality is \((-鈭�, -4)\)

Step by step solution

01

Determine the Critical Points

The critical points are derived by setting the numerator and denominator of the inequality to zero. Therefore: (1) For \(x+3=0\), we have \(x=-3\)(2) For \(x+4=0\), we have \(x=-4\)
02

Test the intervals

Create intervals based on the critical points, which are \(< -4\), \(-4 < x < -3\), and \(> -3\). Take a test point in each interval and replace it in the inequality. Interval 1: Take \(x=-5\), then the expression becomes \(\frac{-5+3}{-5+4}>0\), which is \(\frac{-2}{-1}>0\) that is \(2>0\), hence solution exists in this interval.Interval 2: Take \(x=-3.5\), then the expression becomes \(\frac{-3.5+3}{-3.5+4}<0\), which is \(\frac{-0.5}{0.5}<0\) that is \(-1<0\), hence no solution in this interval.Interval 3: Take \(x=0\), then the expression becomes \(\frac{0+3}{0+4}<0\), which is \(\frac{3}{4}<0\), but this is not true. So, there is no solution in this interval.
03

Graph and Write in Interval Notation

On the number line, plot open dots at the critical points -4 and -3, as the inequality is strictly less than zero (not equal to zero). Draw a line to the left from -4 because the first interval has solutions. There should be no line between -4 and -3 or to the right of -3 as these intervals have no solutions. In interval notation, the solution set is \((-鈭�, -4)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Critical Points in Inequalities
To solve rational inequalities, it's essential to first identify the critical points. Critical points are values which make the expression equal to zero or undefined. In the context of our exercise, \(\frac{x+3}{x+4}<0\), the critical points are found by setting the numerator \(x + 3 = 0\) and the denominator \(x + 4 = 0\) to zero. This yields two values, \(x = -3\) and \(x = -4\), respectively.

These points are significant because they split the number line into distinct intervals. Each interval represents a region we investigate to determine where the inequality holds true. In simple terms, critical points help us \

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Most popular questions from this chapter

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