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Inequalities and absolute value

How to describe a range of numbers precisely: inequality signs, interval notation, and absolute value as distance on the number line.

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Absolute value

Learn the definition of absolute value as distance on the number line, apply the product and quotient rules to simplify expressions, and verify the triangle inequality for specific values.

$|{-7}| = -(-7) = 7$

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Lesson 04 · Absolute value

0,0

x = - 1

a = 4

| x - a |

| x - a | = | - 1 - 4 | = 5

Drag the blue point

x

and the red point

a

along the number line. Drag

a

0

to see

| x |

as the distance from

x

to zero.

This is the diagram from the lesson itself, running here. Drag it - the lesson is built round the thing it shows, not round a picture of it.

The 5 lessons

The 5 lessons in this unit. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.

Reading inequalities

8 questions

Learn how to read and interpret inequality symbols ( $<$ , $>$ , $\leq$ , $\geq$ ), translate everyday language into inequality notation, and determine whether values satisfy compound inequalities like $a < x \leq b$ .

Covers

Inequality symbols and statements

Compound inequalities

Solving linear inequalities

12 questions

Learn how to solve linear inequalities by isolating the variable, applying the sign-flip rule when multiplying or dividing by a negative number.

Covers

Solving by isolating the variable

The sign flip rule

Solving compound inequalities

Interval notation

8 questions

Learn how to read and write interval notation for bounded and unbounded subsets of the number line, and convert fluently between inequality and interval forms.

Covers

Bounded intervals

Unbounded intervals

Absolute value

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12 questions

Learn the definition of absolute value as distance on the number line, apply the product and quotient rules to simplify expressions, and verify the triangle inequality for specific values.

Covers

Definition of absolute value

Product and quotient rules

The triangle inequality

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Absolute value equations and inequalities

8 questions

Learn how to solve absolute value equations by splitting into cases, and absolute value inequalities by converting to compound inequalities expressed in interval notation.

Covers

Solving absolute value equations

Solving absolute value inequalities

Where this unit sits in the curriculum

This unit is part of our Mathematical Foundations for ML learning path, which contains 83 lessons. Every one of them is drawn below.

Nothing has to come first - Inequalities and absolute value is a place to start.

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5 lessons in this unit

78 other lessons in the learning path - after this unit, or alongside it

Every dot is a lesson, every line a prerequisite. You can only start a lesson once you have mastered all of its prerequisites, so you are always building on solid foundations.

Leads to: Derivatives, Functions, Logic, Quadratic equations, Set operations, Trigonometry

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More in Mathematical foundations: Quadratic equations, Set operations, Exponents and roots, Functions, Summation, product, and indexed notation, Logic, Exponentials and logarithms, Trigonometry