Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. (6-x)/(x+2)>1

Verified step by step guidance

1. Start by rewriting the inequality to have zero on one side: subtract 1 from both sides to get \(\frac{6 - x}{x + 2} - 1 > 0\).
2. Combine the terms into a single rational expression by finding a common denominator: \(\frac{6 - x}{x + 2} - \frac{x + 2}{x + 2} > 0\), which simplifies to \(\frac{6 - x - (x + 2)}{x + 2} > 0\).
3. Simplify the numerator: \$6 - x - x - 2 = 4 - 2x\(, so the inequality becomes \)\frac{4 - 2x}{x + 2} > 0$.
4. Identify the critical points by setting numerator and denominator equal to zero: numerator \$4 - 2x = 0\( gives \)x = 2\(, denominator \)x + 2 = 0\( gives \)x = -2$. These points divide the number line into intervals to test.
5. Test values from each interval in the inequality \(\frac{4 - 2x}{x + 2} > 0\) to determine where the expression is positive, then write the solution set in interval notation, excluding points where the denominator is zero.

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 are values 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.