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

Verified step by step guidance

1. Start by writing 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: expand and combine like terms to get a single expression in terms of \(x\).
5. Determine the critical points by setting the numerator and denominator equal to zero separately, then analyze the sign of the expression on intervals defined by these points to find where the inequality holds. Remember to exclude values that make the denominator zero.

Key Concepts

Solving Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. To solve them, first find a common denominator, rewrite the inequality as a single rational expression, and determine where the expression is positive, negative, or zero. Critical points come from zeros of the numerator and denominator, which divide the number line into test intervals.

Domain Restrictions

When solving inequalities with variables in denominators, it is essential to identify values that make the denominator zero, as these are excluded from the domain. These points create vertical asymptotes and must be considered when determining the solution set, ensuring no division by zero occurs.

Interval Notation

Interval notation is a concise way to represent solution sets on the number line. It uses parentheses () to exclude endpoints and brackets [] to include them. For inequalities, intervals show where the solution holds true, combining multiple intervals with unions if necessary.