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| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| Absolute Value | Obsidian::STEM | algebra::abs |
|
Overview
Let x \in \mathbb{R}. The absolute value of x, denoted \lvert x \rvert, is defined as \lvert x \rvert = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x \leq 0 \end{cases}
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Basic
How is the absolute value of x \in \mathbb{R} denoted?
Back: \lvert x \rvert
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
How is the absolute value of x \in \mathbb{R} defined?
Back: \lvert x \rvert = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x \leq 0 \end{cases}
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
The absolute value of x \in \mathbb{R} considers what two cases?
Back: Whether x \geq 0 or x \leq 0.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let x \in \mathbb{R}. When is -\lvert x \rvert \leq x < \lvert x \rvert?
Back: When x < 0.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let x \in \mathbb{R}. When is -\lvert x \rvert < x \leq \lvert x \rvert?
Back: When x > 0.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let x \in \mathbb{R}. When is -\lvert x \rvert \leq x \leq \lvert x \rvert?
Back: Always.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let x, a \in \mathbb{R} and a \geq 0. How is \lvert x \rvert \leq a equivalently written as a chain of inequalities?
Back: -a \leq x \leq a
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let x, a \in \mathbb{R} and a \geq 0. How is \lvert x \rvert \leq a geometricaly depicted?
Back:
!
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let x, a \in \mathbb{R} and a \geq 0. How is -a \leq x \leq a equivalently written using absolute value?
Back: \lvert x \rvert \leq a
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let x, a \in \mathbb{R} and a \geq 0. How is -a \leq x \leq a geometrically depicted?
Back:
!
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Triangle Inequality
Let x, y \in \mathbb{R}. Then the triangle inequality of \mathbb{R} states \lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert$$
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Basic
What does the triangle inequality of \mathbb{R} state?
Back: For x, y \in \mathbb{R}, \lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Why is the triangle inequality named the way it is?
Back: The length of a triangle side is \leq the sum of the other two sides.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
What algebraic inequality is demonstrated in the following?
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Back: The triangle inequality of \mathbb{R}.
Reference: “Triangle Inequality.” In Wikipedia, July 1, 2024. https://en.wikipedia.org/w/index.php?title=Triangle_inequality.
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Basic
What degenerate triangle justifies use of \leq over < in the triangle inequality of \mathbb{R}?
Back:
!
Reference: “Triangle Inequality.” In Wikipedia, July 1, 2024. https://en.wikipedia.org/w/index.php?title=Triangle_inequality.
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Basic
What two chains of inequalities can be added together to prove the triangle inequality of \mathbb{R}?
Back: -\lvert x \rvert \leq x \leq \lvert x \rvert and -\lvert y \rvert \leq y \leq \lvert y \rvert.
Reference: “Triangle Inequality.” In Wikipedia, July 1, 2024. https://en.wikipedia.org/w/index.php?title=Triangle_inequality.
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Basic
What does the general triangle inequality of \mathbb{R} state?
Back: For real numbers a_1, \ldots, a_n, \left\lvert \sum_{k=1}^n a_k \right\rvert \leq \sum_{k=1}^n \lvert a_k \rvert$$
Reference: “Triangle Inequality.” In Wikipedia, July 1, 2024. https://en.wikipedia.org/w/index.php?title=Triangle_inequality.
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Basic
Let a_1\, \ldots, a_n \in \mathbb{R}. What is the following a generalization of? \left\lvert \sum_{k=1}^n a_k \right\rvert \leq \sum_{k=1}^n \lvert a_k \rvert$$
Back: The triangle inequality of \mathbb{R}.
Reference: “Triangle Inequality.” In Wikipedia, July 1, 2024. https://en.wikipedia.org/w/index.php?title=Triangle_inequality.
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Bibliography
- Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
- “Triangle Inequality.” In Wikipedia, July 1, 2024. https://en.wikipedia.org/w/index.php?title=Triangle_inequality.