Solve each rational inequality. Give the solution set in interval notation. 8 /(x - 2) ≥ 2

Verified step by step guidance

1. Start by rewriting the inequality: \(\frac{8}{x - 2} \geq 2\).
2. Bring all terms to one side to have zero on the other side: \(\frac{8}{x - 2} - 2 \geq 0\).
3. Find a common denominator and combine the terms into a single rational expression: \(\frac{8 - 2(x - 2)}{x - 2} \geq 0\).
4. Simplify the numerator: \$8 - 2(x - 2) = 8 - 2x + 4 = 12 - 2x$, so the inequality becomes \(\frac{12 - 2x}{x - 2} \geq 0\).
5. Determine the critical points by setting numerator and denominator equal to zero: numerator \$12 - 2x = 0\( and denominator \)x - 2 = 0$. These points divide the number line into intervals to test the sign of the expression and find the solution set.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where variables appear in the denominator. Solving them requires finding values that satisfy the inequality while ensuring the denominator is not zero, as division by zero is undefined.

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals zero. These points divide the number line into intervals, which are tested to determine where the inequality holds true, helping to identify the solution set.

Interval Notation

Interval notation expresses solution sets as intervals on the number line, using parentheses for excluded endpoints and brackets for included ones. It provides a concise way to represent all values satisfying the inequality.