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Chapter 2: Problem 27

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$9 x^{2}-6 x+1<0$$

Short Answer

Expert verified
\(\emptyset\). The inequality cannot be satisfied for any real number.

Step by step solution

01

Factor the Polynomial

Factor the given polynomial as \((3x-1)^2\). The formula used here is \(a^2-2ab+b^2=(a-b)^2\). Factoring offers a clear view of the structure of the polynomial.
02

Set the Factored Form to Zero

Setting the factored form to zero, i.e., \((3x-1)^2 = 0\), provides the borders of the intervals to be tested. Solving this equation we find that x = 1/3. This splits the number line into two intervals: (-鈭�, 1/3) and (1/3, 鈭�).
03

Test the Intervals

Choose representative numbers from each interval and substitute them into the original inequality. For the interval (-鈭�, 1/3), a good choice is 0. Substituting 0 into the inequality gives \(1 < 0\), which is false. For the interval (1/3, 鈭�), a good choice is 1; substituting 1 gives \(4 < 0\), which is also false.
04

Express the Solution Set

Since the inequality does not satisfy any of the ranges, the solution set to this inequality is an empty set, represented as \(\emptyset\).

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