Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. 4/(3+x)≤3/(3+x)

Verified step by step guidance

1. Step 1: Start by subtracting \( \frac{3}{3+x} \) from both sides of the inequality: \( \frac{4}{3+x} - \frac{3}{3+x} \leq 0 \).
2. Step 2: Combine the fractions on the left side: \( \frac{4 - 3}{3+x} \leq 0 \), which simplifies to \( \frac{1}{3+x} \leq 0 \).
3. Step 3: Analyze the inequality \( \frac{1}{3+x} \leq 0 \). The fraction is negative or zero when the numerator is zero or the denominator is negative.
4. Step 4: Set the denominator \( 3+x \) less than zero: \( 3+x < 0 \). Solve for \( x \) to find \( x < -3 \).
5. Step 5: Since the numerator is always positive (1), the inequality holds when \( x < -3 \). Therefore, the solution set in interval notation is \( (-\infty, -3) \).

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that contain rational functions, which are ratios of polynomials. To solve these inequalities, one must determine where the rational expression is greater than, less than, or equal to a certain value. This often requires finding critical points where the expression is undefined or equals zero, and then testing intervals to establish the solution set.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval (a, b] includes all numbers greater than 'a' and up to 'b', including 'b' itself, which is crucial for clearly expressing the solution set of inequalities.

Critical Points

Critical points are values of the variable where the rational expression is either zero or undefined. These points are essential in solving rational inequalities as they divide the number line into intervals. By testing these intervals, one can determine where the inequality holds true, allowing for the identification of the solution set in interval notation.