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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 17
Chapter 4, Problem 17

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. 5x≤2−3x25x≤2−3x^2

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Rewrite the inequality to have zero on one side by adding \(3x^2\) to both sides: \(5x + 3x^2 \leq 2\).
Bring all terms to one side to set the inequality to zero: \(3x^2 + 5x - 2 \leq 0\).
Factor the quadratic expression \(3x^2 + 5x - 2\) if possible, or use the quadratic formula to find its roots.
Determine the intervals defined by the roots and test a value from each interval in the inequality \(3x^2 + 5x - 2 \leq 0\) to see where it holds true.
Express the solution set as an interval or union of intervals based on the test results and graph this solution on the real number line.

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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 another value using inequality symbols (>, <, ≥, ≤). 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 and Finding Critical Points

To solve polynomial inequalities, first rewrite the inequality so one side is zero, then factor the polynomial if possible. The roots or zeros of the polynomial, called critical points, divide the number line into intervals where the polynomial's sign can be tested.
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Interval Notation and Graphing Solution Sets

After determining where the polynomial is positive or negative, express the solution set using interval notation, which concisely describes all values satisfying the inequality. Graphing on a number line visually represents these intervals, showing included or excluded endpoints.
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Related Practice
Textbook Question

In Exercises 15–18, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] <IMAGE>

f(x)=(x−3)2f(x)=(x−3)^2

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Textbook Question
Show that f(x) = x^3 - 2x - 1 has a real zero between 1 and 2.
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. 4x2 + 1 ≥ 4x

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Textbook Question

Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as z and the difference between y and w.

Textbook Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=(x−4)2−1

Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As x→−2+, f(x)→x\(\to\)-2^{+},\(\text{ }\)f(x)\(\to\) __

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