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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 9
Chapter 4, Problem 9

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. x2−6x+9<0

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1
Recognize that the inequality is a quadratic inequality: \(x^2 - 6x + 9 < 0\).
Factor the quadratic expression on the left side. Notice that \(x^2 - 6x + 9\) is a perfect square trinomial, so it factors as \((x - 3)^2\).
Rewrite the inequality using the factorization: \((x - 3)^2 < 0\).
Consider the properties of squares: since \((x - 3)^2\) is always greater than or equal to zero for all real \(x\), it can never be less than zero.
Conclude that there are no real values of \(x\) that satisfy \((x - 3)^2 < 0\), so the solution set is the empty set, which in interval notation is \(\emptyset\).

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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 symbols like <, >, ≤, or ≥. 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.
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Factoring Quadratic Expressions

Factoring is the process of rewriting a quadratic expression as a product of two binomials. For example, x² - 6x + 9 factors to (x - 3)(x - 3). Factoring helps identify the roots of the polynomial, which are critical points for determining where the inequality changes sign.
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Interval Notation and Number Line Graphing

Interval notation is a concise way to represent sets of real numbers, especially solution sets of inequalities. Graphing on a number line visually shows where the polynomial is positive or negative. Together, they help communicate the solution clearly, indicating which intervals satisfy the inequality.
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Related Practice
Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.


As x→−3−x\(\to\)-3^{-}, f(x)→f\(\left\)(x\(\right\))\(\to\)____

Textbook 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. x2+5x+4>0

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Textbook Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2(x−3)2+1

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Textbook Question

Divide using long division. State the quotient, and the remainder, r(x)r(x). 2x3+7x2+9x−20x+3\(\frac{2x^3 + 7x^2 + 9x - 20}{x + 3}\)

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Textbook Question

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as x and z. y = 25 when x = 2 and z = 5. Find y when x = 8 and z = 12.

Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x1/3 −4x2+7