Solve each rational inequality. Give the solution set in interval notation. (2x-3)(3x+8)/(x-6)³≥0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero separately. Solve for \(x\) in \$2x - 3 = 0\(, \)3x + 8 = 0\(, and \)x - 6 = 0$ to find the values where the expression is zero or undefined.
2. List the critical points found: these points divide the number line into intervals. These points are where the expression can change sign or is undefined.
3. Determine the sign of the expression \(\frac{(2x-3)(3x+8)}{(x-6)^3}\) on each interval by choosing a test point from each interval and substituting it into the expression.
4. Consider the behavior at each critical point: if the factor in the numerator is zero, the expression equals zero there; if the denominator is zero, the expression is undefined and that point is excluded from the solution set.
5. Combine the intervals where the expression is greater than or equal to zero, taking into account the points where the expression equals zero (include these points) and excluding points where the expression is undefined, then write the solution set in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares the expression to zero or another value. Solving them requires finding where the expression is positive, negative, or zero, considering the domain restrictions caused by denominators.

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals zero, dividing the number line into intervals. By testing points in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Interval Notation and Domain Restrictions

Interval notation expresses solution sets compactly using parentheses and brackets to indicate open or closed intervals. Domain restrictions exclude values that make the denominator zero, ensuring the solution set only includes valid inputs for the rational expression.