Solve each rational inequality. Give the solution set in interval notation. (6-x)/(x+2)>1

Verified step by step guidance

1. Start by rewriting the inequality: \(\frac{6 - x}{x + 2} > 1\).
2. Bring all terms to one side to have zero on the other side: \(\frac{6 - x}{x + 2} - 1 > 0\).
3. Find a common denominator and combine the terms: \(\frac{6 - x}{x + 2} - \frac{x + 2}{x + 2} > 0\), which simplifies to \(\frac{6 - x - (x + 2)}{x + 2} > 0\).
4. Simplify the numerator: \$6 - x - x - 2 = 4 - 2x\(, so the inequality becomes \)\frac{4 - 2x}{x + 2} > 0$.
5. Determine the critical points by setting numerator and denominator equal to zero: numerator \$4 - 2x = 0\( and denominator \)x + 2 = 0$. These points divide the number line into intervals to test the sign of the expression.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one rational expression is compared to another using inequality symbols. Solving them requires finding values of the variable that make the inequality true, often by analyzing the sign of the numerator and denominator separately.

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.

Interval Notation

Interval notation is a concise way to express 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.