Problem 44 - 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 to the inequality is \(x \in (2, \infty)\).

Step 1: Factor the Inequality

Let's start by factoring the inequality \(\frac{x+5}{x-2}>0\). The factors we get are \((x + 5)\) and \((x - 2)\). These will be the factors used in formulating the critical points.

Step 2: Identify Critical Points

Set each factor equal to zero and solve for x: \((x + 5) = 0\) gives \(x = -5\) and \((x - 2) = 0\) gives \(x = 2\). The critical points are \(x = -5\) and \(x = 2\). These points divide the real number line into three intervals: \((-\infty, -5)\), \((-5, 2)\), and \((2, \infty)\).

Step 3: Test the Intervals

Choose a number from each interval and replace x in the original inequality and evaluate whether it is true or not: For \((-\infty, -5)\), choose -6: \(\frac{(-6)+5}{(-6)-2}>0\), simplifies to -1 which isn't greater than 0. For \((-5, 2)\), choose 0: \(\frac{(0)+5}{(0)-2}>0\), simplifies to -2.5 which isn't greater than 0. For \((2, \infty)\), choose 3: \(\frac{(3)+5}{(3)-2}>0\), simplifies to 8 which is greater than 0. Therefore, the solution lies within the \((2, \infty)\) interval.

Step 4: Graphing the Solution Set

On the real number line, open circle on 2 and draw a line to the right indicating that x is greater than 2.

Step 5: Express the Solution in Interval Notation

The interval notation for the solution is \((2, \infty)\). This notation indicates that the solution includes all numbers greater than 2 but not including 2 itself. Therefore, the solution to the inequality is \(x \in (2, \infty)\).