Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5.
$\frac{5}{x+3}\ge \frac{3}{x}$

Verified step by step guidance

1. Start by writing the inequality clearly: \(\frac{5}{x + 3} \geq \frac{3}{x}\).
2. Bring all terms to one side to have a single rational expression: \(\frac{5}{x + 3} - \frac{3}{x} \geq 0\).
3. Find a common denominator, which is \(x(x + 3)\), and combine the fractions: \(\frac{5x - 3(x + 3)}{x(x + 3)} \geq 0\).
4. Simplify the numerator: \$5x - 3x - 9 = 2x - 9$, so the inequality becomes \(\frac{2x - 9}{x(x + 3)} \geq 0\).
5. Determine the critical points by setting numerator and denominator equal to zero: \$2x - 9 = 0\(, \)x = 0\(, and \)x + 3 = 0$. Use these points to test intervals on the number line and find where the expression is greater than or equal to zero, then express the solution in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. Solving them requires finding values of the variable that make the inequality true, while ensuring the denominator is never zero to avoid undefined expressions.

Finding a Common Denominator and Combining Terms

To solve rational inequalities, rewrite both sides with a common denominator to combine terms into a single rational expression. This allows comparison to zero and simplifies the inequality into a form suitable for analysis.

Sign Analysis and Interval Notation

After simplifying, determine where the rational expression is positive, negative, or zero by analyzing critical points (zeros and undefined points). Use this to identify solution intervals and express the solution set clearly in interval notation.