Solve each inequality. Give the solution set in interval notation. 4/(x+6)>2/(x-1)

Verified step by step guidance

1. Start by writing down the inequality: \(\frac{4}{x+6} > \frac{2}{x-1}\).
2. To solve the inequality, first bring all terms to one side to have zero on the other side: \(\frac{4}{x+6} - \frac{2}{x-1} > 0\).
3. Find a common denominator for the fractions, which is \((x+6)(x-1)\), and rewrite the inequality as a single fraction: \(\frac{4(x-1) - 2(x+6)}{(x+6)(x-1)} > 0\).
4. Simplify the numerator: expand and combine like terms to get \(\frac{4x - 4 - 2x - 12}{(x+6)(x-1)} > 0\), which simplifies to \(\frac{2x - 16}{(x+6)(x-1)} > 0\).
5. Determine the critical points by setting the numerator and denominator equal to zero: numerator \(2x - 16 = 0\) gives \(x=8\), denominator factors \(x+6=0\) gives \(x=-6\), and \(x-1=0\) gives \(x=1\). These points divide the number line into intervals to test for the inequality.

Key Concepts

Solving Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. To solve them, first bring all terms to one side to form a single rational expression, then determine where this expression is positive or negative by analyzing its sign. Critical points come from zeros of the numerator and denominator, which divide the number line into intervals to test.

Domain Restrictions

When solving inequalities with variables in denominators, it is essential to identify values that make denominators zero, as these are excluded from the domain. These restrictions create boundaries that split the number line and must be considered when expressing the solution set, ensuring no division by zero occurs.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities. It uses parentheses for values not included (open intervals) and brackets for included values (closed intervals). Understanding how to write solutions in interval notation helps clearly communicate the range of values satisfying the inequality.