[tex]$\frac{1}{x-4} \geq 0$[/tex]

Analyze the inequality x − 4 1 ​ ≥ 0 .
Determine that the denominator must be positive: 0"> x − 4 > 0 .
Solve for x : 4"> x > 4 .
Express the solution in interval notation: ( 4 , ∞ ) ​ .

Explanation

Understanding the Inequality We are given the inequality x − 4 1 ​ ≥ 0 . Our goal is to find all values of x that satisfy this inequality.

Analyzing the Denominator The numerator of the fraction is 1, which is positive. Therefore, the fraction x − 4 1 ​ will be greater than or equal to 0 if and only if the denominator x − 4 is positive. Note that x − 4 cannot be equal to 0, because then the fraction would be undefined.

Solving for x We need to solve the inequality 0"> x − 4 > 0 . Adding 4 to both sides, we get 4"> x > 4 .

Expressing the Solution The solution to the inequality is 4"> x > 4 . In interval notation, this is ( 4 , ∞ ) .


Examples
Imagine you're baking a cake and the recipe requires a certain ratio of flour to sugar. If the amount of sugar is represented by x − 4 , and you need to ensure that the ratio x − 4 1 ​ is non-negative to avoid a bad-tasting cake, then 4"> x > 4 tells you the minimum amount of sugar you need to add. This type of inequality helps in real-world scenarios where maintaining a positive ratio or proportion is crucial.

Answered by GinnyAnswer | 2025-07-03

The inequality x − 4 1 ​ ≥ 0 is satisfied for values of x greater than 4. Therefore, the solution is expressed in interval notation as ( 4 , ∞ ) .
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Answered by Anonymous | 2025-07-04