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. x/(x−3)>0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero separately. For the numerator: \(x = 0\). For the denominator: \(x - 3 = 0\), so \(x = 3\). These points divide the number line into intervals to test.
2. Determine the intervals created by the critical points: \((-\infty, 0)\), \((0, 3)\), and \((3, \infty)\). These intervals will be tested to see where the inequality \(\frac{x}{x-3} > 0\) holds true.
3. Choose a test point from each interval and substitute it into the expression \(\frac{x}{x-3}\). Check the sign (positive or negative) of the result to determine if the inequality is satisfied in that interval.
4. Remember that the inequality is strict (\(>\) 0), so exclude points where the expression is zero or undefined. Specifically, exclude \(x = 0\) (where numerator is zero) and \(x = 3\) (where denominator is zero and expression is undefined).
5. Combine the intervals where the expression is positive to write the solution set in interval notation, and then graph these intervals on the real number line, marking excluded points appropriately.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the expression is positive, negative, or zero by analyzing the signs of numerator and denominator.

Critical Points and Sign Analysis

Critical points occur where the numerator or denominator equals zero, dividing the number line into intervals. By testing values in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Interval Notation and Graphing Solutions

Interval notation concisely represents sets of real numbers that satisfy the inequality, using parentheses or brackets to indicate whether endpoints are included. Graphing on a number line visually shows these solution intervals and excluded points, aiding interpretation.