Problem 46 - Algebra and Trigonometry

Question

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+5}{x+2}<0 $$

Step-by-Step Solution

Answer

The solution set for the given inequality in interval notation is \((-5,-2)\)

Step 1: Identify critical values

Critical values are values that make the inequality equals to zero or undefined. Set \(x+5=0\), we get \(x=-5\). This is the value where the inequality equals to zero. Set \(x+2=0\), we get \(x=-2\). This is the value where the inequality is undefined because we cannot divide by zero.

Step 2: Determine the sign of each interval

We have three intervals to consider, which are \((-∞,-5)\), \((-5,-2)\) and \((-2,∞)\). Choose a test point from each interval and substitute it into the given inequality to determine if the inequality holds. If the inequality is true, the interval is part of the solution set, if not, it isn't. The test points can be \(x=-6, -4\) and \(0\). Substituting these values gives \(1/4, -1\) and \(5/2\) respectively. For \(x=-6\), \(\frac{-6+5}{-6+2} = \frac{-1}{-4} = 1/4 > 0\). This is not true for the given inequality, so \((-∞,-5)\) is not part of the solution set. For \(x=-4\), \(\frac{-4+5}{-4+2} = \frac{1}{-2} = -1/2 < 0\). This is true for the given inequality, so \((-5,-2)\) is part of the solution set. For \(x=0\), \(\frac{0+5}{0+2} = \frac{5}{2} > 0\). This is not true for the given inequality, so \((-2,∞)\) is not part of the solution set.

Step 3: Write the solution in interval notation

The solution set is only the interval \((-5,-2)\) because it is the only interval that makes the inequality true. Therefore, the solution set in interval notation is \((-5,-2)\)

Step 4: Graph the solution on a real number line

A number line is used to represent the solution set. The values -2 and -5 are plotted and the region between them is shaded to represent the solution. Note that the end points are open circles because the inequality is '<' rather than '≤'