Solve each rational inequality. Give the solution set in interval notation. (1-x)/(x+2)<-1

Verified step by step guidance

1. Start by rewriting the inequality: \(\frac{1 - x}{x + 2} > -1\).
2. Bring all terms to one side to have zero on the other side: \(\frac{1 - x}{x + 2} + 1 > 0\).
3. Find a common denominator and combine the terms: \(\frac{1 - x}{x + 2} + \frac{x + 2}{x + 2} > 0\), which simplifies to \(\frac{1 - x + x + 2}{x + 2} > 0\).
4. Simplify the numerator: \(\frac{3}{x + 2} > 0\).
5. Determine where the rational expression \(\frac{3}{x + 2}\) is greater than zero by analyzing the sign of the denominator \(x + 2\) (since the numerator 3 is always positive). Also, exclude values that make the denominator zero.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to a number. Solving them requires finding values of the variable that make the inequality true, considering where the expression is defined and the sign of the numerator and denominator.

Critical Points and Sign Analysis

Critical points occur where the numerator or denominator equals zero, dividing the number line into intervals. By testing values in each interval, you determine where the rational expression is positive or negative, which helps identify the solution set for the inequality.

Interval Notation

Interval notation is a concise way to express sets of numbers that satisfy inequalities. It uses parentheses for values not included (like points where the denominator is zero) and brackets for included endpoints, clearly showing the solution set on the number line.