Before you can solve −4x+5>17 you need to be fluent with a small pile of symbols and pictures. This page builds every one of them from absolute zero, in an order where each rests on the one before. If a symbol scared you in the parent note, it will not by the end of this page.
Everything in this topic lives on one horizontal line of numbers. Before we talk about "greater" or "less", we must agree on what those words look like.
The single most important habit for this whole topic:
This is why −5 is smaller than −2: even though "5" looks bigger than "2", the point −5 sits further left. Look at the figure — −5 is genuinely to the left of −2. Hold onto this; it is the secret behind the flip rule later.
The topic needs the number line because a linear inequality's answer is never one number — it is a stretch of the line. We need a picture that can show a whole stretch at once. This connects to Number Line and Real Numbers.
WHY four symbols and not one? Because two different questions live here:
Which side? — that is < versus > (direction).
Is the boundary itself allowed? — that is the little bar under ≤ / ≥ (inclusion).
The extra bar = glued underneath is doing real work: it says "the boundary point counts too." Keep those two questions separate in your head; almost every mistake in this topic is confusing one for the other.
Recall Quick check: which symbol?
Fill the blank: −7□−3.
Answer ::: −7<−3, because −7 sits further left.
Contrast this with Linear Equations, where "solve" usually gives you one lit-up point (e.g. x=4). An inequality lights up a whole stretch. That difference is the entire personality of this topic.
We require a=0. WHY? If a=0 then ax+b=b: the x has vanished, there is nothing left to solve. So a=0 is the promise that "this really is a problem about x." Curved cousins live at Quadratic Inequalities.
This is the concept the flip rule secretly depends on, so we build it carefully with a picture.
Watch what the mirror does to order, in the figure:
Before the flip: 2 is left of 5, so 2<5.
After the flip: −2 landed to the right of −5, so now −2>−5.
The order got reversed by the mirror. Nothing magical — reflection literally swaps left and right, and "left/right" is how we defined smaller/larger. This single picture is the honest reason the inequality sign flips whenever you multiply or divide by a negative number.
Read it top to bottom: the number line and the sign idea are the bedrock; they support the symbols and the flip rule; those combine with the variable and the linear form to produce a solution set, which we finally draw (circles) and write (intervals).
