What is the solution set of the quadratic inequality [tex]x^2-5 \leq 0[/tex]?
A. {x | -5 \leq x \leq \sqrt{5}}
B. {x | -5 \leq x \leq 5}
C. {x | -\sqrt{5} \leq x \leq 5}
D. {x | -\sqrt{5} \leq x \leq \sqrt{5}}

Rewrite the inequality: x 2 ≤ 5 .
Take the square root of both sides: − 5 ​ ≤ x ≤ 5 ​ .
Express the solution set in set notation.
The solution set is { x ∣ − 5 ​ ≤ x ≤ 5 ​ } ​ .

Explanation

Understanding the Inequality We are given the quadratic inequality x 2 − 5 ≤ 0 . Our goal is to find all values of x that satisfy this inequality.

Rewriting the Inequality First, we can rewrite the inequality as x 2 ≤ 5 . This means we are looking for values of x whose square is less than or equal to 5.

Taking the Square Root To solve this, we take the square root of both sides. Remember to consider both positive and negative square roots, since squaring a negative number also results in a positive number. This gives us − 5 ​ ≤ x ≤ 5 ​ .

Expressing the Solution Set Therefore, the solution set is all x such that x is greater than or equal to − 5 ​ and less than or equal to 5 ​ . In set notation, this is written as { x ∣ − 5 ​ ≤ x ≤ 5 ​ } .


Examples
Imagine you are designing a square garden, and you want its area to be no more than 5 square meters. If x represents the length of one side of the garden, then the area is x 2 . The inequality x 2 ≤ 5 tells you that the side length x must be between − 5 ​ and 5 ​ . Since side lengths cannot be negative, x must be between 0 and 5 ​ meters. This problem demonstrates how quadratic inequalities can be used to model real-world constraints.

Answered by GinnyAnswer | 2025-07-08

The solution set of the quadratic inequality x 2 − 5 ≤ 0 is { x ∣ − 5 ​ ≤ x ≤ 5 ​ } , which is option D.
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Answered by Anonymous | 2025-08-18