Problem 51 - 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. $$ \frac{x}{x-3}>0 $$

Step-by-Step Solution

Answer

The solution set for the inequality \(\frac{x}{x-3}>0\) is \((3, \infty)\). On the number line, it is represented by an open circle at 3 with all points to the right shaded.

Step 1: Factor the inequality

The given inequality is already factored, represented as \(\frac{x}{x-3}>0\). We see the rational function is undefined at \(x=3\).

Step 2: Identify the critical points

The critical points are obtained by setting the numerator and the denominator equal to zero. From the numerator, we obtain \(x=0\) and from the denominator we get \(x=3\). So, our critical points are 0 and 3.

Step 3: Test intervals

We divide the number line into intervals using the critical points and test the sign of our function in each interval to determine whether the inequality is satisfied. Our intervals are \(-\infty\) to 0, 0 to 3, and 3 to \(\infty\). If we take a test point -1 in the first interval, substitute it into inequality we get negative, hence \(\frac{x}{x-3}<0\) for \(x<0\). If we take a test point 1 in the second interval, substitute it into inequality we get negative, hence \(\frac{x}{x-3}<0\) for \(00\) for \(x>3\).

Step 4: Write the solution in interval notation and draw a graph on the number line

Since our inequality is \(\frac{x}{x-3}>0\), we consider only the intervals where the function was positive. Thus, our solution set is \(x>3\), which in interval notation is \((3, \infty)\). We draw a graph on the number line, making an open circle at 3 and shading everything to the right of it.