Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Linear Inequalities

College Algebra

1. Equations & Inequalities

Linear Inequalities

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1:56 minutes

Problem 91

Textbook Question

Solve each inequality in Exercises 86–91 using a graphing utility.1/(x + 1) ≤ 2/(x + 4)

Verified step by step guidance

Rearrange the inequality to have a single rational expression on one side: \( \frac{1}{x + 1} - \frac{2}{x + 4} \leq 0 \).

Find a common denominator for the rational expressions, which is \((x + 1)(x + 4)\).

Rewrite each fraction with the common denominator: \( \frac{(x + 4) - 2(x + 1)}{(x + 1)(x + 4)} \leq 0 \).

Simplify the numerator: \( (x + 4) - 2(x + 1) = x + 4 - 2x - 2 = -x + 2 \).

Set the simplified inequality: \( \frac{-x + 2}{(x + 1)(x + 4)} \leq 0 \) and use a graphing utility to find the intervals where this inequality holds true.

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Key Concepts

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

Inequalities

Inequalities express a relationship where one quantity is less than, greater than, or equal to another. In this case, the inequality involves a rational expression, which requires understanding how to manipulate and solve inequalities involving fractions. The solution set may include intervals or specific values that satisfy the inequality.

Rational Functions

A rational function is a ratio of two polynomials. In the given inequality, the expressions 1/(x + 1) and 2/(x + 4) are rational functions. Understanding their behavior, such as asymptotes and intercepts, is crucial for graphing and solving the inequality effectively.

Graphing Utilities

Graphing utilities are tools that allow for the visualization of functions and inequalities. They can help identify where the graphs of the two sides of the inequality intersect and where one is less than or equal to the other. Using these tools effectively can simplify the process of finding solutions to inequalities.