Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. (2x - 1)(5x - 9)(x - 4) < 0

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First, identify the critical points by setting each factor equal to zero: solve $2x - 1 = 0$, $5x - 9 = 0$, and $x - 4 = 0$.

Next, solve each equation to find the zeros: $x = \frac{1}{2}$, $x = \frac{9}{5}$, and $x = 4$. These points divide the number line into four intervals.

Determine the sign of the product $(2x - 1)(5x - 9)(x - 4)$ on each interval by choosing a test point from each interval and substituting it into the inequality.

Analyze the sign of each factor at the test points and multiply the signs to find whether the product is positive or negative on that interval.

Finally, select the intervals where the product is less than zero (negative) and express the solution set in interval notation, excluding the points where the product equals zero.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality signs (<, >, ≤, ≥). Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Critical Points and Sign Analysis

Critical points are the values of the variable that make each factor of the polynomial zero. These points divide the number line into intervals. By testing a value from each interval in the inequality, you determine whether the polynomial is positive or negative there, helping to identify solution intervals.

Interval Notation

Interval notation is a concise way to represent sets of real numbers between two endpoints. Parentheses () indicate that endpoints are not included, while brackets [] mean they are included. It is used to express the solution set of inequalities clearly and efficiently.

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