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Ch. 1 - Equations and Inequalities
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 1 - Equations and InequalitiesProblem 70
Chapter 2, Problem 70

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

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1
Start by rewriting the inequality: \(\frac{1}{x+2} \geq 3\).
Bring all terms to one side to have zero on the other side: \(\frac{1}{x+2} - 3 \geq 0\).
Find a common denominator and combine the terms into a single rational expression: \(\frac{1 - 3(x+2)}{x+2} \geq 0\).
Simplify the numerator: \(\frac{1 - 3x - 6}{x+2} = \frac{-3x - 5}{x+2} \geq 0\).
Determine the critical points by setting numerator and denominator equal to zero: solve \(-3x - 5 = 0\) and \(x + 2 = 0\), then analyze the sign of the 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 variables appear in the denominator. Solving them requires finding values that satisfy the inequality while ensuring the denominator is not zero, as division by zero is undefined.
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Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals zero, dividing the number line into intervals. Testing each interval helps determine where the inequality holds true by analyzing the sign of the expression.
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Point-Slope Form

Interval Notation

Interval notation expresses solution sets using brackets and parentheses to indicate inclusive or exclusive endpoints. It concisely represents all values satisfying the inequality, excluding points where the expression is undefined.
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Interval Notation
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