Problem 60 - 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. $$ \frac{x}{x+2} \geq 2 $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x > -2\) and in interval notation is \((-2, \infty)\)

Step 1: Rearranging the Inequality

Rearrange the inequality to set it to zero on one side. This will simplify the inequality to \( \frac{x}{x+2} - 2 \geq 0 \). With the help of common denominator of \(x+2\), this can further be written as \( \frac{x - 2(x+2)}{x+2} \geq 0 \)

Step 2: Simplifying the Expression

Simplify the numerator of the left side of the inequality to get \( \frac{-x-4}{x+2} \geq 0 \)

Step 3: Find the Critical Values

Find the x-values that make the inequality equal to zero. This is given by the roots of the numerator and the denominator of the simplified rational expression. In this case, \( x = -2 \) from \( x + 2 = 0 \) is the denominator root but it is not considered because a denominator cannot be zero, and \( x = -4 \) from \( -x - 4 = 0 \) is the numerator root. Thus, the critical values are \( x = -4 \)

Step 4: Check the Intervals

The critical values divide the number line into intervals. Test any value within these intervals into the original inequality. The intervals are \( x < -4 \), \( -4 < x < -2 \), and \( x > -2 \). If the x values chosen in these intervals satisfy the inequality, then all x values in the interval will satisfy the inequality.

Step 5: Solution Set and Interval Notation

The solution to the inequality is the x values in the intervals that satisfied the original inequality. Therefore, the solution is \( x > -2 \). The corresponding solution set in interval notation is \((-2, \infty)\).