1. What an inequality says
An equation ($=$) says two expressions describe the same number. An inequality says they don't, and how they relate.
| Symbol | Read as | Solutions look like |
|---|---|---|
| $x < a$ | "strictly less than" | Everything to the left of $a$ on the number line, not including $a$. |
| $x > a$ | "strictly greater than" | Everything to the right of $a$, not including $a$. |
| $x \leq a$ | "less than or equal to" | Everything to the left, including $a$. |
| $x \geq a$ | "greater than or equal to" | Everything to the right, including $a$. |
The solution set of an inequality is the set of all values of the variable that make it true. Where the solution set of $x + 3 = 10$ is a single number ($x = 7$), the solution set of $x + 3 < 10$ is a whole region of the number line ($x < 7$ — every number less than $7$, with $7$ itself excluded).
The symbols $<$ and $>$ are strict — they exclude equality. $\leq$ and $\geq$ include it. On a number-line graph this difference shows up as an open versus filled circle at the endpoint, which we'll see in §5.
2. Solving linear inequalities
Solving a linear inequality is almost identical to solving a linear equation. You isolate the variable using the same toolkit:
- Add or subtract the same number on both sides.
- Multiply or divide both sides by the same positive number.
- Simplify each side.
Solve $3x + 5 < 17$:
$$ 3x + 5 < 17 \quad\Longrightarrow\quad 3x < 12 \quad\Longrightarrow\quad x < 4. $$The solution set is "all $x$ less than $4$" — infinitely many numbers, with $4$ itself excluded.
Check by trying a value inside the set: at $x = 0$, $3(0) + 5 = 5$, and $5 < 17$ ✓. Try a value outside: at $x = 5$, $3(5) + 5 = 20$, and $20 < 17$ is false ✓.
3. The flip rule for negatives
Here is the one place inequalities differ from equations:
If you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality.
Why? Multiplying by a negative reflects across zero, and reflection swaps "left" and "right." If $3 < 5$, then negating both gives $-3$ and $-5$ — and now $-3 > -5$, because $-3$ is to the right of $-5$ on the number line. The relationship has flipped.
Concrete example: solve $-2x \geq 8$.
Divide both sides by $-2$ — and flip $\geq$ to $\leq$:
$$ -2x \geq 8 \quad\Longrightarrow\quad x \leq -4. $$Check: $x = -5$ should be in the solution set. $-2 \cdot (-5) = 10$, and $10 \geq 8$ ✓.
Forgetting to flip when multiplying or dividing by a negative. Whenever the coefficient of $x$ comes out negative, take a moment to write the flip explicitly. If you stayed inside the positives all the way through, no flip is needed.
Adding or subtracting a negative does not require flipping — the rule is strictly about multiplication and division. $x - 3 < 5$ becomes $x < 8$ without any flipping, even though "$-3$" is involved.
4. Compound inequalities
Two inequalities joined into one statement. There are two flavors, "AND" and "OR," and they describe different regions.
AND (intersection)
$2 < x < 7$ means $x > 2$ and $x < 7$ — both must be true. The solution set is the strip between $2$ and $7$, exclusive of the endpoints.
To solve compound AND inequalities written as a "sandwich," do the same operation to all three pieces. $-1 < 2x - 3 \leq 5$:
Add $3$ throughout: $2 < 2x \leq 8$.
Divide by $2$ throughout: $1 < x \leq 4$.
OR (union)
$x < -1$ or $x \geq 4$ is satisfied by either condition — values to the left of $-1$, or values at or to the right of $4$. The solution set is two disjoint pieces of the number line.
OR inequalities are written as two separate inequalities joined by the word "or." There's no compact sandwich notation; each piece is solved independently.
An AND like $x > 5$ AND $x < 2$ has no solutions — there's no number bigger than $5$ and smaller than $2$. The solution set is the empty set, $\emptyset$. OR compounds, by contrast, almost always cover something.
5. Graphing solutions on the number line
The standard convention:
- Filled (closed) circle at the endpoint: the endpoint is included ($\leq$, $\geq$).
- Open circle at the endpoint: the endpoint is not included ($<$, $>$).
- Shaded line: the set of solutions extending from the endpoint(s).
- Arrow at the end of the shaded region: the solution extends to infinity in that direction.
For a compound inequality like $1 < x \leq 4$, you'd draw an open circle at $1$, a closed circle at $4$, and shade the segment between.
Solution sets can also be written in interval notation, which is a compact textual form of the same idea: parentheses for excluded endpoints, brackets for included ones. So $x < 4$ is $(-\infty, 4)$; $x \geq 4$ is $[4, \infty)$; $1 < x \leq 4$ is $(1, 4]$. Infinity always gets a parenthesis — it's not a number to include.