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Chapter 3: Problem 28

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ 4 x^{2}-4 x+1 \geq 0 $$

Short Answer

Expert verified
The root of the equation is \(x = 0.5\). Because the leading coefficient of the polynomial is positive and the graph opens upwards, the entire real line is a solution to the inequality. Therefore, the solution to the inequality \(4x^2 - 4x + 1 \geq 0\) is \(-\infty, +\infty\)

Step by step solution

01

Simplify the inequality

First, the inequality is simplified by combining similar terms. In this case, the inequality should be rewritten as follows: \(4x^2 - 4x + 1 \geq 0\).
02

Find the roots

Use the quadratic formula to find the roots. The quadratic formula is \(-b卤\sqrt{b^2-4ac}/2a\), where \(a = 4\), \(b = -4\), and \(c = 1\). Plug these values into the formula and calculate the roots.
03

Inspect the intervals

The roots split the number line into intervals. Check the sign of the polynomial on each interval.
04

Solution set

State the solution set in interval notation.

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