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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1:27 minutes

Problem 106

Textbook Question

Answer the following. Why must -4 be in the solution set of x+4 / 2x+1 ≥ 0? (Do not solve the inequality.)

Verified step by step guidance

Identify the expression given in the inequality: \( \frac{x+4}{2x+1} \geq 0 \).

Consider the critical points where the expression is undefined or equal to zero. These occur when the numerator \( x+4 = 0 \) and the denominator \( 2x+1 = 0 \).

Solve for \( x \) in the numerator: \( x+4 = 0 \) gives \( x = -4 \). This is a critical point where the expression equals zero.

Solve for \( x \) in the denominator: \( 2x+1 = 0 \) gives \( x = -\frac{1}{2} \). This is a critical point where the expression is undefined.

Since \( x = -4 \) makes the numerator zero, it is a point where the expression equals zero, thus satisfying the inequality \( \geq 0 \). Therefore, \( x = -4 \) must be included in the solution set.

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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 side is not necessarily equal to the other, often using symbols like '≥', '≤', '>', or '<'. Understanding how to manipulate and interpret inequalities is crucial for determining solution sets, as they indicate ranges of values that satisfy the given condition.

Critical Points

Critical points are values of the variable where the expression changes its sign, typically found by setting the numerator and denominator of a rational expression to zero. In the context of the inequality x + 4 / (2x + 1) ≥ 0, identifying these points helps determine where the expression is positive or negative, which is essential for understanding the solution set.

Solution Set

The solution set of an inequality includes all values of the variable that satisfy the inequality condition. For the given inequality, understanding why -4 must be included in the solution set involves analyzing the behavior of the expression at critical points and determining the intervals where the inequality holds true.