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>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 \( x = 0 \). These points divide the number line into intervals.
2. The critical points are \( x = -4 \) and \( x = 0 \). Use these points to split the real number line into three intervals: \( (-\infty, -4) \), \( (-4, 0) \), and \( (0, \infty) \).
3. Determine the sign of the rational expression \( \frac{x+4}{x} \) on each interval by choosing a test point from each interval and substituting it into the expression.
4. Analyze the inequality \( \frac{x+4}{x} > 0 \) by selecting intervals where the expression is positive based on the sign test results.
5. Express the solution set in interval notation, remembering to exclude points where the expression is undefined (denominator zero) and to consider whether to include points where the numerator is zero based on the strict inequality.

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 are values that make the numerator or denominator zero, dividing the number line into intervals. By testing points 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 inequalities, using parentheses or brackets to indicate open or closed intervals. Graphing on a number line visually shows these solution sets, highlighting excluded points like zeros of the denominator.