Final answer:
To solve the inequality (√(1-4x^2) - 1) / x ≥ -4, first set the expression equal to -4 and solve for x. Simplify the equation and solve for x^2. Since a square root of a negative number is not defined in the real number system, the solution is x = 0.00139. Therefore, the correct answer is option d) x ≥ -3/4.
Step-by-step explanation:
To solve the inequality (√(1-4x^2) - 1) / x ≥ -4, we need to consider different cases. First, let's set the expression equal to -4 and solve for x:
(√(1-4x^2) - 1) / x = -4
(√(1-4x^2) - 1) = -4x
Now, square both sides of the equation:
1-4x^2 - 2√(1-4x^2) + 1 = 16x^2
Combine like terms:
20x^2 - 2√(1-4x^2) = 0
Isolate the radical term:
2√(1-4x^2) = 20x^2
Square both sides again:
4(1-4x^2) = 400x^4
Simplify:
4 - 16x^2 = 400x^4
Rearrange the equation:
400x^4 + 16x^2 - 4 = 0
This is a quadratic equation in terms of x^2. Solving it gives us two solutions:
x^2 = -0.0024 or x^2 = 0.00139
Since a square root of a negative number is not defined in the real number system, the solution is x = 0.00139. Therefore, the correct answer is option d) x ≥ -3/4.