Answer the following. Why must -4 be in the solution set of $\frac{x+4}{2x+1}\ge 0$? (Do not solve the inequality.)

Verified step by step guidance

1. First, identify the expression given: \(\frac{x+4}{2x+1} \geq 0\).
2. Understand that the solution set includes values of \(x\) for which the expression is greater than or equal to zero.
3. Notice that the numerator is \(x+4\). When \(x = -4\), the numerator becomes zero, making the entire fraction equal to zero.
4. Since the inequality includes 'greater than or equal to zero', the value of \(x\) that makes the fraction exactly zero must be included in the solution set.
5. Therefore, \(x = -4\) must be in the solution set because it makes the fraction equal to zero, satisfying the inequality.

Key Concepts

Inequality and Solution Set

An inequality compares expressions and defines a range of values (solution set) that satisfy it. Understanding which values make the inequality true or false is essential to determine the solution set without necessarily solving it fully.

Critical Points and Undefined Values

Critical points occur where the numerator or denominator of a rational expression is zero. Values that make the denominator zero are excluded from the solution set because they make the expression undefined, while zeros of the numerator often belong to the solution set.

Sign Analysis of Rational Expressions

Sign analysis involves determining where a rational expression is positive, negative, or zero by examining the signs of numerator and denominator separately. This helps identify intervals where the inequality holds true and explains why certain points must be included in the solution.