Question

Solve each rational inequality. Give the solution set in interval notation. 3/(x-2)\u003c1

Accepted Answer

Start by rewriting the inequality $$ \frac{3}{x-2} \u003c 1  to $$have zero on one side. Subtract 1 from both sides to get $$ \frac{3}{x-2} - 1 \u003c 0 . $$Find a common denominator to combine the terms on the left side: $$ \frac{3}{x-2} - \frac{x-2}{x-2} \u003c 0 . $$This simplifies to $$ \frac{3 - (x-2)}{x-2} \u003c 0 . $$Simplify the numerator: $$ 3 - (x - 2) = 3 - x + 2 = 5 - x . So $$the inequality becomes $$ \frac{5 - x}{x - 2} \u003c 0 . $$Determine the critical points by setting the numerator and denominator equal to zero: $$ 5 - x = 0 $$ gives $$ x = 5 $$, and $$ x - 2 = 0 $$ gives $$ x = 2 . $$These points divide the number line into intervals to test. Test values from each interval around the critical points $$ x = 2 $$ and $$ x = 5  in $$the inequality $$ \frac{5 - x}{x - 2} \u003c 0  to $$determine where the expression is negative. Remember to exclude $$ x = 2 $$ from the solution set because it makes the denominator zero.

Solve each rational inequality. Give the solution set in interval notation. 3/(x-2)<1

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Key Concepts

Rational Inequalities

Domain Restrictions

Interval Notation and Sign Analysis