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4.7: Quadratic Functions
https://math.libretexts.org/Courses/Highline_College/Math_084__Intermediate_Algebra_Foundations_for_Soc_Sci_Lib_Arts_and_GenEd/04%3A_Functions/4.07%3A_Quadratic_Functions
You may recall studying quadratic equations in Intermediate Algebra. In this section, we review those equations in the context of our next family of functions: the quadratic functions.
6.2: Solving Absolute Value Equations
https://math.libretexts.org/Bookshelves/Applied_Mathematics/Calculus_for_Business_and_Social_Sciences_Corequisite_Workbook_(Dominguez_Martinez_and_Saykali)/06%3A_Absolute_Value/6.02%3A_Solving_Absolute_Value_Equations
To solve absolute value equations, first consider the following two properties of Absolute Value. It is important to check the solutions by substituting them back into the original equation. Finally, ...To solve absolute value equations, first consider the following two properties of Absolute Value. It is important to check the solutions by substituting them back into the original equation. Finally, the solution set of an absolute value equation is typically graphed as points on a number line.
1.2: Negative Numbers
https://math.libretexts.org/Courses/Northeast_Wisconsin_Technical_College/College_Technical_Math_1A_(NWTC)/01%3A_Operations_with_Real_Numbers/1.02%3A_Negative_Numbers
Negative numbers are a fact of life, from winter temperatures to our bank accounts. Let’s practice evaluating expressions involving negative numbers.
3.4: Using Cases in Proofs
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/03%3A_Constructing_and_Writing_Proofs_in_Mathematics/3.04%3A_Using_Cases_in_Proofs
In the case where $x \ge 0$ , we see that $|x| = x$ , and since $|x| = a$ , we can conclude that $x = -a$ . In the case where $x \ge 0$ , we know that $|x| = x$ and so the inequality \(|x| < a\...In the case where $x \ge 0$ , we see that $|x| = x$ , and since $|x| = a$ , we can conclude that $x = -a$ . In the case where $x \ge 0$ , we know that $|x| = x$ and so the inequality $|x| < a$ implies that $x < a$ . So in both cases, we have proven that $-a < x < a$ and this proves that if $|x| < a$ , then $-a < x < a$ . In Part (1) of Theorem 3.25, we proved that for each real number $x$ , $|x| < a$ if and only if $-a < x < a$ .
4.9: Absolute Value Functions
https://math.libretexts.org/Courses/Highline_College/Math_084__Intermediate_Algebra_Foundations_for_Soc_Sci_Lib_Arts_and_GenEd/04%3A_Functions/4.09%3A_Absolute_Value_Functions
There are a few ways to describe what is meant by the absolute value |x| of a real number x. The long and short of both of these procedures is that |x| takes negative real numbers and assigns them t...There are a few ways to describe what is meant by the absolute value |x| of a real number x. The long and short of both of these procedures is that |x| takes negative real numbers and assigns them to their positive counterparts while it leaves positive numbers alone. This last description is the one we shall adopt, and is summarized and discuss in this Module.
3.4: Using Cases in Proofs
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/03%3A_Constructing_and_Writing_Proofs_in_Mathematics/3.04%3A_Using_Cases_in_Proofs
In the case where $x \ge 0$ , we see that $|x| = x$ , and since $|x| = a$ , we can conclude that $x = -a$ . In the case where $x \ge 0$ , we know that $|x| = x$ and so the inequality \(|x| < a\...In the case where $x \ge 0$ , we see that $|x| = x$ , and since $|x| = a$ , we can conclude that $x = -a$ . In the case where $x \ge 0$ , we know that $|x| = x$ and so the inequality $|x| < a$ implies that $x < a$ . So in both cases, we have proven that $-a < x < a$ and this proves that if $|x| < a$ , then $-a < x < a$ . In Part (1) of Theorem 3.25, we proved that for each real number $x$ , $|x| < a$ if and only if $-a < x < a$ .
2.3: Quadratic Functions
https://math.libretexts.org/Courses/Cosumnes_River_College/Math_370%3A_Precalculus/02%3A_Linear_and_Quadratic_Functions/2.03%3A_Quadratic_Functions
You may recall studying quadratic equations in Intermediate Algebra. In this section, we review those equations in the context of our next family of functions: the quadratic functions.
4.4: Absolute Value Equations and Inequalities as Applied to Distance
https://math.libretexts.org/Bookshelves/Precalculus/Corequisite_Companion_to_Precalculus_(Freidenreich)/4%3A_Inequalities/4.04%3A_Absolute_Value_Equations_and_Inequalities_as_Applied_to_Distance
The absolute value function, denoted y = |x|, takes any negative real number input and outputs the positive version of that number. Nonnegative numbers are left unchanged. Measuring distance is a goo...The absolute value function, denoted y = |x|, takes any negative real number input and outputs the positive version of that number. Nonnegative numbers are left unchanged. Measuring distance is a good application to demonstrate the usefulness of this function. Distance is never negative.
2.1: Absolute Value Functions
https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/02%3A_Polynomial_and_Rational_Functions/2.01%3A_Absolute_Value_Functions
This section explores absolute value functions, including their definition, properties, and graphing. It explains how to interpret and solve absolute value equations and inequalities, and covers trans...This section explores absolute value functions, including their definition, properties, and graphing. It explains how to interpret and solve absolute value equations and inequalities, and covers transformations such as shifts and reflections. Examples and exercises help demonstrate how to handle these functions in different contexts, reinforcing the understanding of their behavior and applications.
4.2: Absolute Value
https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/04%3A_Absolute_Value_Functions/4.02%3A_Absolute_Value
The absolute value of a number is a measure of its magnitude, sans (without) its sign.
2.2: Absolute Value Functions
https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_(Stitz-Zeager)_-_Jen_Test_Copy/02%3A_Linear_and_Quadratic_Functions/2.02%3A_Absolute_Value_Functions
There are a few ways to describe what is meant by the absolute value |x| of a real number x. The long and short of both of these procedures is that |x| takes negative real numbers and assigns them t...There are a few ways to describe what is meant by the absolute value |x| of a real number x. The long and short of both of these procedures is that |x| takes negative real numbers and assigns them to their positive counterparts while it leaves positive numbers alone. This last description is the one we shall adopt, and is summarized and discuss in this Module.

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