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Chapter 2: Problem 35

Explain what is unusual about the solution set of the inequality. $$4 x^{2}-4 x+1 \leq 0$$

Short Answer

Expert verified
The unusual aspect of this inequality is that it has only one solution, not an interval: \(x = 0.5\).

Step by step solution

01

Recognize the quadratic form

The inequality is presented as \(4x^{2}-4x+1\), which is the form of a perfect square, and it can be rewritten as \( (2x-1)^2\).
02

Simplify the inequality

So, the inequality \(4x^{2} - 4x + 1 \leq 0\) could be rewritten as \((2x-1)^2 \leq 0\).
03

Solve the inequality

Since the left side of the inequality is a square, it will always be greater or equal to 0. However, a square can only be equal to 0 when the expression within the bracket is 0. We can solve \(2x-1=0\) to get the solution x=0.5. So, there is a single solution for this inequality.
04

Conclusion about the solution set

The unusual feature to be noted is that this quadratic inequality has a single solution point, which is \(x = 0.5\), instead of an interval or a range. This is because the square of a real number is always non-negative. Therefore, \( (2x-1)^2\) can only be zero, but never less than zero.

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