4a²x⁴ ≥ x² × √(1 - 16r² + 4)
Answer (2)
To solve the inequality 4 a 2 x 4 ≥ x 2 × 1 − 16 r 2 + 4 , we'll break it down step-by-step.
Simplify the Right Side:
First, simplify the expression inside the square root: 1 − 16 r 2 + 4 .
Combine the constants inside: 1 + 4 = 5 .
So, the expression becomes 5 − 16 r 2 .
Rewrite the Inequality:
Substitute back into the original inequality: 4 a 2 x 4 ≥ x 2 × 5 − 16 r 2 .
Divide Both Sides by x 2 (assuming x = 0 to avoid division by zero):
4 a 2 x 2 ≥ 5 − 16 r 2 .
Consider the Domains:
The expression 5 − 16 r 2 is defined only when 5 − 16 r 2 ≥ 0 , which simplifies to r 2 ≤ 16 5 .
Ensure x = 0 as we divided by x 2 .
Solve for x :
Without more context or specific values, we resolve as it stands since inequality cannot be simplified further algebraically without specific values for a , x , and r .
Thus, we've simplified and analyzed the inequality, and any further numeric solutions would require specific values or constraints for a , x , and r . This inequality represents a conditions-based scenario in algebra where the solution set depends on the parameters' values and constraints.
To solve the inequality, we first simplify the right side to \text{sqrt}(5 - 16r^2) and then rewrite it as 4a^2x^2 \text{sqrt}(5 - 16r^2) . The solution depends on values for a , x , and r that satisfy the constructed inequality and the condition derived.
[section_start id=explanation]
To solve the inequality 4a^2x^4 \u2265 x^2 \times \text{sqrt}(1 - 16r^2 + 4) , we will break it down step-by-step.
Simplify the Right Side:
First, simplify the expression inside the square root: \text{sqrt}(1 - 16r^2 + 4) . Combine the constants inside: 1 + 4 = 5 . Thus, the expression becomes \text{sqrt}(5 - 16r^2) .
Rewrite the Inequality:
Insert the simplified expression back into the inequality: 4a^2x^4 x^2 \times \text{sqrt}(5 - 16r^2) .
Divide Both Sides by x^2 (assuming x 0 to avoid division by zero):
After division, we have 4a^2x^2 \text{sqrt}(5 - 16r^2) .
Consider the Domains:
Next, for \text{sqrt}(5 - 16r^2) to be defined, we require 5 - 16r^2 0 , which simplifies to r^2 \frac{5}{16} . Remember to ensure x 0 as this was a condition for our division.
**Analyze and Solve for x : **
This inequality currently cannot be simplified further without specific values for a , x , and r . To find valid solutions, we need to consider what values satisfy 4a^2x^2 \text{sqrt}(5 - 16r^2) , based on the range for r derived above.
In conclusion, we have simplified the inequality conditionally. Further numeric or graphical solutions could involve substituting specific values for a , x , and r as required, taking care over the conditions derived from the square root.
A practical example could involve assigning values such as a = 1 , r = 0.25 , and solving for x to check if the condition holds true.
The understanding of square roots and inequalities from algebra shows the necessity of non-negative expressions within roots, emphasizing its application in solving inequalities.
