Solve each rational inequality. Give the solution set in interval notation. -4/(1-x)<5

Verified step by step guidance

1. Start by rewriting the inequality clearly: \(\frac{-4}{1 - x} < 5\).
2. Bring all terms to one side to have zero on the other side: \(\frac{-4}{1 - x} - 5 < 0\).
3. Find a common denominator to combine the terms into a single rational expression: \(\frac{-4 - 5(1 - x)}{1 - x} < 0\).
4. Simplify the numerator: \(-4 - 5 + 5x = 5x - 9\), so the inequality becomes \(\frac{5x - 9}{1 - x} < 0\).
5. Determine the critical points by setting numerator and denominator equal to zero: \$5x - 9 = 0\( and \)1 - x = 0$. These points divide the number line into intervals to test the sign of the expression and find where it is less than zero.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one side is a ratio of polynomials. Solving them requires finding values of the variable that make the inequality true, considering where the expression is defined and where it changes sign.

Domain Restrictions

Since rational expressions have denominators, values that make the denominator zero are excluded from the solution set. Identifying these restrictions is crucial to avoid undefined expressions and to correctly determine the solution intervals.

Interval Testing and Sign Analysis

After finding critical points (where numerator or denominator is zero), the number line is divided into intervals. Testing values from each interval helps determine where the inequality holds true, allowing the solution set to be expressed in interval notation.