Inequality Basics

The four inequality symbols

An inequality compares two quantities that need not be equal. In place of the equals sign it uses one of four symbols.

So $5 < 8$ says five is less than eight, and $8 > 5$ says eight is greater than five. Those are the same fact read from opposite ends. The symbol always opens toward the larger amount and narrows to a point at the smaller one. So the pointed end aims at the smaller number: in $5 < 8$ the point sits on the $5$.

That two-ended reading is worth stating as a rule, because it lets you turn any inequality around. Take $7 > x$: swap the two sides and reverse the symbol, and you get $x < 7$, the same fact read from the other end. You will use this often to move a variable to the left. In general, the statements $a \le b$ and $b \ge a$ say exactly the same thing: ”$a$ is at most $b$” is the same claim as ”$b$ is at least $a$.”

The two symbols with a bar underneath, $\le$ and $\ge$, allow equality as well as the strict comparison. The statement $x \le 6$ is true when $x$ is $6$ and also true when $x$ is anything below $6$. We call $<$ and $>$ strict inequalities, because they exclude the boundary, and $\le$ and $\ge$ inclusive (or non-strict), because they let it in. That one bar is the whole difference between admitting the boundary number and shutting it out.

Every number that makes an inequality true is part of its solution set. Ask which numbers satisfy $x > 3$: the value $4$ works, and so do $5$, $3.1$, $3.0001$, and a million. Every number to the right of $3$ qualifies, so this solution set holds infinitely many numbers. That is why we describe the answer as a range and draw it, rather than listing it.

Picturing the solution on a number line

A number line turns a solution set into a picture you can take in at a glance. Locate the boundary number from the inequality, decide whether the boundary itself belongs, and shade the direction that holds the rest of the solutions. Three decisions capture every case.

The circle at the boundary records whether the boundary is a solution. Draw an open (hollow) circle for a strict inequality, $<$ or $>$, because the boundary is excluded. Draw a filled (closed) circle for an inclusive inequality, $\le$ or $\ge$, because the boundary is included. The shaded ray then runs in the direction of the solutions: to the right for a “greater” relation ($>$ or $\ge$), toward the larger numbers. The ray runs to the left for a “less” relation ($<$ or $\le$). The arrowhead means the solutions run on forever that way.

Here is the graph of $x > 2$. The circle on $2$ is open, because $2$ is not a solution, and the shading runs right across every number larger than $2$.

The only thing that changes for $x \ge 2$ is the circle: it becomes filled, because now $2$ belongs to the solution. Switching between a strict and an inclusive relation never moves the shading, it only opens or closes the boundary circle.

That last sentence is the one to test rather than take on trust, and the figure below is built for it. Its three controls are exactly the three decisions described above. “Endpoint” sets where the boundary sits, “Side” picks which way to shade, and “Endpoint is” switches the boundary in or out.

Set “Endpoint” to $2$ and “Side” to right, and you have the graph of $x > 2$ from the picture above. Now switch “Endpoint is” between out and in, and watch the whole figure. The ray does not move a pixel, the sign in the readout changes from $>$ to $\ge$, and the circle fills in. One decision, one thing on the screen, and it is the decision students most often lose. Then set “Endpoint” to $-1$, “Side” to left, and “Endpoint is” to in: that is the graph the next worked example asks for before you read it.

Interval notation

Drawing a picture every time is slow, so there is a compact written shorthand for the same solution set, called interval notation. An interval is written as its left and right bounds inside a pair of fences, with the kind of fence recording whether each bound is included.

A square bracket includes the endpoint, matching a filled circle and an inclusive symbol. A parenthesis excludes the endpoint, matching an open circle and a strict symbol. So $2 \le x \le 5$ becomes $[2, 5]$, while $2 < x < 5$ becomes $(2, 5)$. The two can be mixed: $2 \le x < 5$ is the half-open interval $[2, 5)$, closed on the left and open on the right.

When the solution runs on forever in one direction, we mark that end with the infinity symbol, $\infty$ for the right and $-\infty$ for the left. Here is the one firm rule: infinity always takes a parenthesis, never a bracket. Infinity is not a number, so there is no endpoint to reach or to include; the parenthesis records that the interval is open and unbounded. That is why $x > 2$ is written $(2, \infty)$ and $x \le -1$ is written $(-\infty, -1]$, with a bracket on the real endpoint and a parenthesis at infinity. The table lines up all three ways of saying the same thing.

One caution about reading. Written on its own, a symbol pair like $(2, 5)$ could be an interval on a line or a point $(x, y)$ in the plane from the last chapter. The surrounding words tell you which is meant: an interval names a set of numbers on one axis, while a coordinate point names one location in the plane.

The rules for changing an inequality

To rearrange an inequality without breaking it, you need to know which moves keep the statement true. Three of the rules match your equation habits exactly, and one carries a twist. In each case the reason is visible on the number line, where $a < b$ simply means $a$ sits to the left of $b$.

Adding or subtracting the same quantity from both sides keeps the direction. Sliding both numbers the same distance in the same direction does not change which one is on the left. Starting from the true statement $3 < 7$ and adding $4$ to each side gives $7 < 11$, still true, and subtracting $10$ from each side of $3 < 7$ gives $-7 < -3$, also true. The order survives every shift.

Multiplying or dividing both sides by a positive number keeps the direction. Scaling by a positive factor stretches or shrinks every distance from zero but never carries a number across zero, so the left-right order holds. From $3 < 7$, multiplying both sides by $2$ gives $6 < 14$, and dividing $6 < 14$ by $2$ returns $3 < 7$. Both stay true.

Multiplying or dividing both sides by a negative number reverses the direction. This is the twist, and it is the single most error-prone fact about inequalities, so it deserves a careful look. Take the true statement $3 < 5$ and multiply both sides by $-1$. The left becomes $-3$ and the right becomes $-5$. If the symbol stayed put we would be claiming $-3 < -5$, but that is false. On the number line $-5$ lies to the left of $-3$, so $-3$ is the larger number. The true statement is $-3 > -5$, with the symbol flipped.

Transitivity chains comparisons together. If $a < b$ and $b < c$, then $a < c$. On the line, $a$ is left of $b$ and $b$ is left of $c$, so $a$ must be left of $c$. A quick check: since $2 < 5$ and $5 < 9$, transitivity gives $2 < 9$, which is plainly true. You will use this soon to check whether a chain like $a < x < b$ can describe any numbers at all.

These rules are enough to take a short step on an inequality, the same way you take one step on an equation. For $x + 3 < 7$, subtract $3$ from both sides, and since subtraction never disturbs the symbol, $x < 4$. For $-2x < 6$, divide both sides by $-2$, and because $-2$ is negative you reverse the symbol, giving $x > -3$. Carrying such a problem through several steps in a reliable order is the job of the next lesson. Here the point is only that each individual move is governed by the rules above.

Compound inequalities

Sometimes a value is squeezed from both sides at once, larger than one number and smaller than another. A compound inequality writes both bounds in a single line. The notation $a < x < b$ means $x > a$ and $x < b$ at the same time, so $x$ lies strictly between $a$ and $b$. Read it as two claims joined by “and”: $x$ is above $a$, and $x$ is below $b$.

Both symbols in the chain point the same way: never one “less than” symbol and one “greater than” symbol. Use $<$ or $\le$ throughout, or use $>$ or $\ge$ throughout; that is the syntax. But writing it correctly is not enough to guarantee it describes any numbers. If $x > a$ and $x < b$ are both true, transitivity forces $a < b$: the left bound must be smaller than the right bound. We write $-2 < x \le 3$, which reads left to right as ”$-2$ is less than $x$, and $x$ is at most $3$,” and it makes sense because $-2 < 3$. Now look at $3 < x < -2$: both symbols point the same way, so the syntax looks fine, but the numbers do not cooperate. It asks for a number above $3$ and below $-2$, and since $3$ is not less than $-2$, nothing satisfies both parts at once. Check the numbers, not just the arrows, before trusting a compound inequality.

Each end keeps its own circle: an open circle where the bound is strict and a filled circle where it is inclusive. On the number line the solution is the band between the two boundaries rather than a ray running off to one side. In interval notation the same band is just the pair of bounds in one set of fences, each fence chosen by its own symbol. So $-2 < x \le 3$, open on the left and closed on the right, is the half-open interval $(-2, 3]$.

Common mistakes