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Ch. 1 - Equations and Inequalities
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 1 - Equations and InequalitiesProblem 43
Chapter 2, Problem 43

Solve each quadratic inequality. Give the solution set in interval notation. 2x2-9x≤18

Verified step by step guidance
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Start by rewriting the inequality in standard form by moving all terms to one side: \(2x^{2} - 9x \leq 18\) becomes \(2x^{2} - 9x - 18 \leq 0\).
Next, factor the quadratic expression \(2x^{2} - 9x - 18\). To do this, look for two numbers that multiply to \(2 \times (-18) = -36\) and add to \(-9\).
Once factored, set each factor equal to zero to find the critical points (roots) of the quadratic equation. These points divide the number line into intervals.
Determine the sign of the quadratic expression on each interval by choosing test points from each interval and substituting them back into the factored inequality.
Based on the inequality \(\leq 0\), select the intervals where the quadratic expression is less than or equal to zero, and write the solution set in interval notation including the roots where the expression equals zero.

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Key Concepts

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

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value. Solving it requires finding the range of x-values that satisfy the inequality, often by analyzing the related quadratic equation and its graph.
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Solving Quadratic Equations

To solve a quadratic inequality, first solve the corresponding quadratic equation by setting it equal to zero. Techniques include factoring, completing the square, or using the quadratic formula to find critical points that divide the number line.
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Solving Quadratic Equations by Factoring

Interval Notation and Test Intervals

After finding critical points, the number line is divided into intervals. Test points from each interval in the original inequality to determine where it holds true. The solution set is then expressed in interval notation, showing all x-values that satisfy the inequality.
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Interval Notation
Related Practice
Textbook Question

Solve each problem. See Examples 5 and 6. Formaldehyde is an indoor air pollutant formerly found in plywood, foam insulation, and carpeting. When concentrations in the air reach 33 micrograms per cubic foot (μg/ft3), eye irritation can occur. One square foot of new plywood could emit 140 μg per hr. (Data from A. Hines, Indoor Air Quality & Control.) A room has 100 ft2 of new plywood flooring. Find a linear equation F that computes the amount of formaldehyde, in micrograms, emitted in x hours.

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Textbook Question

Solve each problem. See Examples 5 and 6. Formaldehyde is an indoor air pollutant formerly found in plywood, foam insulation, and carpeting. When concentrations in the air reach 33 micrograms per cubic foot (μg/ft^3), eye irritation can occur. One square foot of new plywood could emit 140 μg per hr. (Data from A. Hines, Indoor Air Quality & Control.) The room contains 800 ft^3 of air and has no ventilation. Determine how long it would take for concentrations to reach 33 μg/ft^3. (Round to the nearest tenth.)

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Solve each equation or inequality. |4x + 3| - 2 = -1

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