Solve each rational inequality. Give the solution set in interval notation. 3/(x-6)≤2

Verified step by step guidance

1. Start by rewriting the inequality: \(\frac{3}{x-6} \leq 2\).
2. Bring all terms to one side to have zero on the other side: \(\frac{3}{x-6} - 2 \leq 0\).
3. Find a common denominator and combine the terms into a single rational expression: \(\frac{3 - 2(x-6)}{x-6} \leq 0\).
4. Simplify the numerator: \$3 - 2x + 12 = 15 - 2x$, so the inequality becomes \(\frac{15 - 2x}{x-6} \leq 0\).
5. Determine the critical points by setting numerator and denominator equal to zero: numerator \$15 - 2x = 0\( and denominator \)x - 6 = 0$. These points divide the number line into intervals to test for the inequality.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where variables appear in the denominator. Solving them requires finding values of the variable that make the inequality true, while considering restrictions where the denominator is zero to avoid undefined expressions.

Finding Critical Points

Critical points are values where the numerator or denominator equals zero. These points divide the number line into intervals to test the inequality. Identifying these points helps determine where the rational expression changes sign.

Interval Testing and Solution Sets

After finding critical points, test values from each interval in the inequality to check if they satisfy it. Combine intervals where the inequality holds true, and express the solution set in interval notation, excluding points that make the denominator zero.