Solve each inequality. Give the solution set using interval notation. 3/x-1 ≤ 5/x+3

Verified step by step guidance

1. Start by rewriting the inequality clearly: \(\frac{3}{x-1} \leq \frac{5}{x+3}\).
2. Bring all terms to one side to have zero on the other side: \(\frac{3}{x-1} - \frac{5}{x+3} \leq 0\).
3. Find a common denominator, which is \((x-1)(x+3)\), and combine the fractions: \(\frac{3(x+3) - 5(x-1)}{(x-1)(x+3)} \leq 0\).
4. Simplify the numerator: \$3(x+3) - 5(x-1) = 3x + 9 - 5x + 5 = -2x + 14$, so the inequality becomes \(\frac{-2x + 14}{(x-1)(x+3)} \leq 0\).
5. Identify critical points by setting numerator and denominator equal to zero: numerator \(-2x + 14 = 0\) gives \(x = 7\), denominator zeros are \(x = 1\) and \(x = -3\). Use these points to test intervals on the number line to determine where the inequality holds.

Key Concepts

Solving Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. To solve them, first find values that make denominators zero (excluded from the domain), then rewrite the inequality to a single rational expression and analyze its sign over intervals determined by critical points.

Finding Critical Points and Domain Restrictions

Critical points are values where the numerator or denominator equals zero, dividing the number line into intervals. Domain restrictions exclude points where the denominator is zero, as the expression is undefined there. These points help determine where the inequality changes sign.

Interval Notation for Solution Sets

Interval notation expresses solution sets as intervals on the number line, using parentheses for excluded endpoints and brackets for included ones. After testing intervals between critical points, write the solution set by combining intervals where the inequality holds true.