By the end of this topic you will solve things like "the absolute value of 2x+5 equals 3x−1" or "the absolute value of x−1 plus the absolute value of x+2 equals 5" — but we will not even write those with bars until §3, once the bars mean something. Below is every symbol and idea the parent note leans on, built from nothing, in an order where each one only uses the ones before it.
Absolutely everything in this topic lives on a single horizontal line of numbers.
Why the topic needs it: absolute value is defined as distance from zero, and "distance" only means something once you have a ruler laid out. The number line is that ruler. See Number line and intervals.
Written as a rule (a two-part rule — one formula for each side of zero; we'll give this kind of rule its proper name "piecewise" in §7):
∣x∣={x−xif x≥0if x<0
If x is already right of zero (x≥0), its distance isx — leave it alone.
If x is left of zero (x<0), then x itself is negative, so we write −x to flip it positive. Example: x=−3 gives ∣−3∣=−(−3)=3.
The parent note claims ∣x∣=x2. Let's earn every symbol.
Now the chain: square x (sign vanishes), then square-root (we get the positive root back). Positive-and-same-size-as-x — that is exactly the distance from zero.
(−3)2=9=3=∣−3∣.
Why the topic mentions it: it ties absolute value to the Distance formula you meet in geometry, and it's a slick way to prove properties without splitting into cases.
Why both matter:∣x∣=5 pins down exact spots (two points); ∣x∣<5 describes a whole stretch of the line; ∣x∣>5 describes two stretches. Reading the sign correctly is the difference between "AND" and "OR" later.
The arrows below mean "is needed before": start at a box with no incoming arrow, and you may only move to a box once you have mastered everything pointing into it. Read it as a checklist of prerequisites flowing downward into the three big skills — equations, inequalities, and the hardest, compound multi-bar problems. Notice that "Absolute value equals distance from zero" (§3) is the hub every branch passes through: get that one idea and everything else is bookkeeping.
Test yourself — cover the right side and answer out loud. Each line below is a mini flashcard: the text before the ::: is the question, and the text after it is the hidden answer to reveal.
What does ∣x∣ mean in plain words?
The distance from x to 0 on the number line; always ≥0.
Evaluate ∣−3∣ using the two-part rule.
x<0 so ∣−3∣=−(−3)=3.
Is −x always negative?
No — it is the opposite of x; if x=−3 then −x=3.
Why does x2=∣x∣?
Squaring kills the sign, the root returns the positive value — that's the distance.
What does A=±b stand for in full?
A=borA=−b — the two-case split.
Why does ∣A∣=b give two answers, not one?
Two points sit exactly b steps from zero: one right (b), one left (−b).
How many solutions does ∣A∣=b have when b<0?
None — a distance can never be negative.
Which joining word goes with ∣A∣<b, AND or OR?
AND (a single "between" stretch): −b<A<b.
Which joining word goes with ∣A∣>b?
OR (two outside stretches) joined by ∪.
What is the symbol ∞ and can it be an endpoint?
It means "goes on forever", is not a number, and never counts as an endpoint (always a round bracket).
What does the bracket in [−7,3] include that (−7,3) does not?
The endpoints −7 and 3 (closed vs open).
What is a critical point of ∣x+2∣ and why does it matter?
x=−2, where the inside is zero and the sign flips; it splits the line into regions.
What is the union symbol ∪ telling you to do?
Combine two separate pieces of the number line into one answer set.
Return here whenever a symbol in the parent topic feels unfamiliar — the whole subject is just distances on the line you built in §1.
