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Chapter 14: Problem 1

When solving a quadratic inequality, how do you know when to include and when to exclude the endpoints in the solution set?

Short Answer

Expert verified
To determine whether to include or exclude endpoints in the solution set of a quadratic inequality, first find the critical points by solving the corresponding quadratic equation: \(ax^2 + bx + c = 0\). If the inequality is greater than (>) or less than (<), exclude the endpoints. If the inequality is greater than or equal to (鈮�) or less than or equal to (鈮�), include the endpoints. In the given exercise, we are solving ax^2 + bx + c > 0 or ax^2 + bx + c < 0, so we exclude the endpoints in the solution set.

Step by step solution

01

Understand quadratic inequalities and the solution set

A quadratic inequality is a mathematical expression in the form of ax^2 + bx + c > 0 or ax^2 + bx + c < 0 (where a, b, and c are constants, and a 鈮� 0). The solution set of a quadratic inequality consists of all the x-values that, when plugged into the inequality, make it true.
02

Find the critical points

To determine if the endpoints should be included or excluded in the solution set, we need to find the critical points of the inequality. These are the points where the inequality equals zero (i.e., ax^2 + bx + c = 0).
03

Solve the corresponding quadratic equation for the critical points

Solve the quadratic equation ax^2 + bx + c = 0 using factoring, completing the square, or the quadratic formula: \((-b \pm \sqrt{b^2 - 4ac})/(2a)\).
04

Analyzing the inequality sign to determine the inclusion or exclusion of endpoints

Once we find the critical points, we can determine whether to include or exclude them in the solution set depending on the inequality sign: 1. If the inequality is greater than (>) or less than (<), the endpoints are excluded from the solution set. 2. If the inequality is greater than or equal to (鈮�) or less than or equal to (鈮�), the endpoints are included in the solution set. In our case, the exercise states that we are solving a quadratic inequality. So, for ax^2 + bx + c > 0 and ax^2 + bx + c < 0, we exclude the endpoints in the solution set.

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Most popular questions from this chapter

Fill in the blank with the correct term. Some of the given choices will not be used. Descartes' rule of signs a vertical asymptote the leading-term test \(\quad\) an oblique the intermediate \(\quad\) asymptote value theorem direct variation the fundamental \(\quad\) inverse variation theorem of algebra a horizontal line a polynomial function a vertical line a rational function \(\quad\) parallel a one-to-one function \(\quad\) perpendicular a constant function a horizontal asymptote If it is possible for graph more than once, then the graph is not the graph of a function.

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Fill in the blank with the correct term. Some of the given choices will not be used. Descartes' rule of signs a vertical asymptote the leading-term test \(\quad\) an oblique the intermediate \(\quad\) asymptote value theorem direct variation the fundamental \(\quad\) inverse variation theorem of algebra a horizontal line a polynomial function a vertical line a rational function \(\quad\) parallel a one-to-one function \(\quad\) perpendicular a constant function a horizontal asymptote Two lines with slopes \(m_{1}\) and \(m_{2}\) are if and only if the product of their slopes is \(-1\)

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