Problem-solving strategies

The number system

Exponent laws

Scientific notation

Units and measurement

Radicals

Absolute value

MYP Standard Mathematics Absolute Value Notes

Absolute Value Measures Distance, Not Direction

Definition

Absolute value

For a real number $x$, the absolute value $|x|$ is its distance from $0$ on the number line.

Absolute value is also called the modulus or numerical value.

Because it represents a distance, it is never negative.

Note

A distance can be $0$ (when you are at the point), but it cannot be negative.
This is why $|x|\ge 0$ for every real number $x$.

Two Equivalent Definitions Give You Two Ways to Work

One standard definition is piecewise (it depends on whether $x$ is negative or not): $$|x|=\begin{cases} -x, & x<0 \\ \phantom{-}x, & x\ge 0 \end{cases}$$

A second equivalent definition is: $$|x|=\sqrt{x^2}$$

These two definitions agree because squaring makes any real number non-negative, and then the square root returns the non-negative value.

Common Mistake

Do not confuse $|x|$ with $-x$.
They are only the same when $x\le 0$.
For example, if $x=5$, then $|x|=5$ but $-x=-5$.

What Numbers Can Be Absolute Values?

Since $|x|\ge 0$, any non-negative real number could be an absolute value.

$\sqrt{25}=5$ can be an absolute value.
$\sqrt{17}$ can be an absolute value (it is positive).
$-\sqrt{2}$ cannot be an absolute value (it is negative).

This leads to an important property: $$|x|\ge 0\text{ for all real }x,\quad \text{and }|x|=0\text{ only when }x=0.$$

Active recall

Can $0$ be an absolute value?
Can $-0.1$ be an absolute value? Explain using the "distance" idea.

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MYP Standard Mathematics Absolute Value Notes

Absolute Value Measures Distance, Not Direction

Two Equivalent Definitions Give You Two Ways to Work

What Numbers Can Be Absolute Values?

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Excepteur sint occaecat cupidatat non proident

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Absolute Value Measures Distance, Not Direction

Two Equivalent Definitions Give You Two Ways to Work

What Numbers Can Be Absolute Values?

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Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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Number and Algebra Foundations7 subtopics

Coordinate Geometry and Linear Modelling5 subtopics

Functions and Quadratic Relationships4 subtopics

Sets, Logic and Probability4 subtopics

Geometry of Shapes, Surface Area and Volume3 subtopics

Statistical Measures and Data Representation3 subtopics

Right Triangles and Circle Geometry3 subtopics

Sequences, Formulae and Proportional Reasoning3 subtopics

Sampling Methods and Bivariate Data2 subtopics

MYP Standard Mathematics Absolute Value Notes

Number and Algebra Foundations7 subtopics

Coordinate Geometry and Linear Modelling5 subtopics

Functions and Quadratic Relationships4 subtopics

Sets, Logic and Probability4 subtopics

Geometry of Shapes, Surface Area and Volume3 subtopics

Statistical Measures and Data Representation3 subtopics

Right Triangles and Circle Geometry3 subtopics

Sequences, Formulae and Proportional Reasoning3 subtopics

Sampling Methods and Bivariate Data2 subtopics

Absolute Value Measures Distance, Not Direction

Two Equivalent Definitions Give You Two Ways to Work

What Numbers Can Be Absolute Values?

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Absolute Value Measures Distance, Not Direction

Two Equivalent Definitions Give You Two Ways to Work

What Numbers Can Be Absolute Values?

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account