Question

Solve each rational inequality. Give the solution set in interval notation. (x+3)/(2x-5)≤1

Accepted Answer

Start by rewriting the inequality $$ \frac{x+3}{2x-5} \leq 1  so $$that one side is zero. Subtract 1 from both sides to get $$ \frac{x+3}{2x-5} - 1 \leq 0 . $$Combine the terms on the left side over a common denominator: $$ \frac{x+3}{2x-5} - \frac{2x-5}{2x-5} \leq 0 . $$This simplifies to $$ \frac{x+3 - (2x-5)}{2x-5} \leq 0 . $$Simplify the numerator: $$ x + 3 - 2x + 5 = -x + 8 . So $$the inequality becomes $$ \frac{-x + 8}{2x - 5} \leq 0 . $$Identify the critical points by setting numerator and denominator equal to zero: numerator zero at $$ -x + 8 = 0 $$ which gives $$ x = 8 $$, and denominator zero at $$ 2x - 5 = 0 $$ which gives $$ x = \frac{5}{2} . $$These points divide the number line into intervals. Test values from each interval in the inequality $$ \frac{-x + 8}{2x - 5} \leq 0  to $$determine where the expression is less than or equal to zero. Remember to exclude values where the denominator is zero, and include points where the numerator is zero if the inequality allows equality.

Solve each rational inequality. Give the solution set in interval notation. (x+3)/(2x-5)≤1

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Key Concepts

Rational Inequalities

Critical Points and Sign Analysis

Interval Notation