Problem 27 - 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. $$ 9 x^{2}-6 x+1<0 $$

Step-by-Step Solution

Answer

The inequality \(9x^{2}-6x+1<0\) has no solutions, represented by the empty set ().

Step 1: Express Polynomial in Standard Form

Start by rewriting the inequality \(9x^{2}-6x+1<0\) in standard form. This form can be factored as \((3x - 1)^{2}<0\).

Step 2: Find the Root of the Quadratic Polynomial

Solve \((3x - 1)^{2}=0\), which gives a root at \(x={1}{3}\)

Step 3: Evaluate Intervals Around the Root

The root divides the number line into two intervals: \(-∞, \frac{1}{3}\) and \( \frac{1}{3}, +∞ \). Test these intervals by picking any sample point in each interval and substitute the points into the inequality. According to the trichotomy law, since \((3x - 1)^{2}\) is a square, it can either be equal to zero or greater than zero. There's no x-value that makes \((3x - 1)^{2}\) to be less than zero.

Step 4: Graph and Express Solution in Interval Notation

Since there's no value of x that makes the inequality less than zero, the inequality does not exist. Therefore, there is no solution for the inequality. Finally, the interval notation representing the solution set is empty, denoted by (). The graph on the real number line is a line with no markings, indicating no solution.