Problem 28 - Algebra and Trigonometry

Question

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. $$ 4 x^{2}-4 x+1 \geq 0 $$

Step-by-Step Solution

Answer

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 1: Simplify the inequality

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

Step 2: Find the roots

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

Step 3: Inspect the intervals

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

Step 4: Solution set

State the solution set in interval notation.