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

Verified step by step guidance

1. First, identify the inequality: $\frac{3}{x-1}\le \frac{5}{x+3}$.
2. To eliminate the fractions, find a common denominator, which is $(x-1)(x+3)$.
3. Multiply both sides of the inequality by the common denominator $(x-1)(x+3)$ to clear the fractions.
4. Simplify the resulting expression and solve the inequality: $3(x+3)\le 5(x-1)$.
5. Solve the inequality for $x$ and express the solution set in interval notation, considering any restrictions from the original inequality.

Key Concepts

Inequalities

Inequalities are mathematical statements that express the relationship between two expressions that are not necessarily equal. They can be represented using symbols such as ≤ (less than or equal to), ≥ (greater than or equal to), < (less than), and > (greater than). Solving inequalities involves finding the values of the variable that make the inequality true, which may include considering the direction of the inequality when multiplying or dividing by negative numbers.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. In the context of inequalities, it is important to identify the values that make the denominator zero, as these values are excluded from the solution set. Understanding how to manipulate and simplify rational expressions is crucial for solving inequalities involving them, as it allows for clearer comparisons and easier identification of solution intervals.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints, using parentheses and brackets to indicate whether the endpoints are included or excluded. For example, (a, b) means all numbers between a and b, excluding a and b, while [a, b] includes both endpoints. This notation is particularly useful for expressing the solution sets of inequalities, as it provides a concise way to communicate the range of values that satisfy the inequality.