Question

Solve each rational inequality. Give the solution set in interval notation. (x - 1)/(x - 4) \u003e 0

Accepted Answer

Identify the critical points by setting the numerator and denominator equal to zero separately: solve $$x - 1 = 0$$ and $$x - 4 = 0. $$These points divide the number line into intervals. The critical points are $$x = 1$$ and $$x = 4. $$These points are where the rational expression can change sign or be undefined (note that $$x = 4$$ makes the denominator zero, so it is excluded from the domain). Determine the sign of the expression $$\frac{x - 1}{x - 4} on $$each interval created by the critical points: $$( -\infty, 1 )$$, $$(1, 4)$$, and $$(4, \infty). $$Choose a test point from each interval and substitute it into the expression to check if the result is positive or negative. Since the inequality is $$\frac{x - 1}{x - 4} \u003e 0$$, select the intervals where the expression is positive based on your test points. Remember to exclude $$x = 4$$ because the expression is undefined there. Express the solution set in interval notation by combining the intervals where the inequality holds true, using parentheses to indicate that endpoints where the expression is zero or undefined are not included.

Solve each rational inequality. Give the solution set in interval notation. (x - 1)/(x - 4) > 0

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Key Concepts

Rational Inequalities

Critical Points and Sign Analysis

Interval Notation