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−4)/(x+3) > 0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero separately: solve \( x - 4 = 0 \) and \( x + 3 = 0 \). These points divide the number line into intervals.
2. Determine the intervals based on the critical points found: \( (-\infty, -3) \), \( (-3, 4) \), and \( (4, \infty) \).
3. Test a sample value from each interval in the inequality \( \frac{x - 4}{x + 3} > 0 \) to check whether the expression is positive or negative in that interval.
4. Remember that the inequality is strict (greater than zero), so exclude points where the expression equals zero or is undefined (i.e., exclude \( x = 4 \) and \( x = -3 \)).
5. Combine the intervals where the inequality holds true and express the solution set in interval notation, then graph these intervals on the real number line.

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, considering the domain restrictions where the denominator is not zero.

Critical Points and Sign Analysis

Critical points are values that make the numerator or denominator 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 on a Number Line

Interval notation concisely represents solution sets using parentheses and brackets to indicate open or closed intervals. Graphing on a number line visually shows these intervals, highlighting where the inequality is satisfied and marking excluded points such as zeros of the denominator.