Solve each inequality. Give the solution set using interval notation.

Verified step by step guidance

1. Rewrite the inequality clearly: \(\frac{5x + 2}{x + 1} < 0\).
2. Identify the critical points by setting the numerator and denominator equal to zero separately: solve \$5x + 2 = 0\( and \)x + 1 = 0$ to find values where the expression is zero or undefined.
3. Use the critical points to divide the number line into intervals. These intervals will be tested to determine where the inequality holds true.
4. Choose a test point from each interval and substitute it into the expression \(\frac{5x + 2}{x + 1}\) to check if the result is less than zero.
5. Based on the test results, write the solution set in interval notation, remembering to exclude points where the denominator is zero (since the expression is undefined there).

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the numerator and denominator change sign and determining intervals where the inequality holds true.

Critical Points and Sign Analysis

Critical points are values that make the numerator or denominator zero, dividing the number line into intervals. By testing points in each interval, you can determine the sign of the rational expression and identify where the inequality is satisfied.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets to indicate open or closed intervals. It clearly shows the range of values that satisfy the inequality, excluding points where the expression is undefined.