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

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Start by rewriting the inequality: \(\frac{(x+2)}{(2x+3)} \leq 5\).

Bring all terms to one side to have zero on the other side: \(\frac{(x+2)}{(2x+3)} - 5 \leq 0\).

Combine the terms over a common denominator: \(\frac{(x+2) - 5(2x+3)}{(2x+3)} \leq 0\).

Simplify the numerator: \(\frac{x + 2 - 10x - 15}{2x + 3} \leq 0\), which simplifies to \(\frac{-9x - 13}{2x + 3} \leq 0\).

Determine the critical points by setting numerator and denominator equal to zero separately: solve \(-9x - 13 = 0\) and \(2x + 3 = 0\), then analyze the sign of the rational expression on intervals defined by these points to find where the inequality holds.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to a number or another expression. Solving them requires finding values of the variable that make the inequality true, considering where the expression is defined and the sign of the numerator and denominator.

Critical Points and Sign Analysis

Critical points occur where the numerator or denominator equals zero, dividing the number line into intervals. By testing values in each interval, you determine where the rational expression satisfies the inequality. This sign analysis helps identify solution intervals while excluding points that make the denominator zero.

Interval Notation

Interval notation is a concise way to express sets of real numbers that satisfy inequalities. It uses parentheses for excluded endpoints and brackets for included endpoints, clearly showing the range of solutions. Proper use of interval notation is essential for communicating the solution set of inequalities.