Solving and Graphing Inequalities

Not every mathematical statement uses an equals sign. Sometimes the relationship between two quantities is that one is greater or smaller than the other, not exactly equal. That's where inequalities come in. An inequality compares two expressions and describes their relative size, not their exact value.

The symbols you'll use are:

$<$ means "less than"
$>$ means "greater than"
$\leq$ means "less than or equal to"
$\geq$ means "greater than or equal to"

When you solve an inequality, you're finding all the values that make the inequality true. Unlike an equation, which usually has one solution, an inequality typically has infinitely many solutions.

Reading Inequality Symbols

Think of the inequality symbol as an alligator's mouth that always wants to eat the bigger number.

The statement $x > 3$ means "$x$ is greater than 3." The mouth is open toward $x$, showing that $x$ is the larger value. So $x$ could be 4, 5, 100, or any number greater than 3.

The statement $x \leq 7$ means "$x$ is less than or equal to 7." Here, $x$ could be 7, 6, 0, -10, or any number that's either equal to 7 or smaller than 7.

Solving Inequalities: Almost Like Equations

Solving inequalities works almost exactly like solving equations. You can add, subtract, multiply, or divide both sides by the same number to isolate the variable. The goal is to get the variable alone on one side.

Example: Solve $x + 5 > 12$.

Subtract 5 from both sides: $$x > 7$$

That's the solution. Any number greater than 7 makes this inequality true. Check: if $x = 8$, then $8 + 5 = 13$, and $13 > 12$ ✓.

Example: Solve $3y \leq 15$.

Divide both sides by 3: $$y \leq 5$$

Any number less than or equal to 5 works. If $y = 5$, then $3(5) = 15$, and $15 \leq 15$ is true (since the symbol includes "equal to").

Example: Solve $\frac{n}{4} > 2$.

Multiply both sides by 4: $$n > 8$$

The Special Rule: Multiplying or Dividing by Negatives

Here's where inequalities differ from equations in an important way. When you multiply or divide both sides by a negative number, you must flip the inequality symbol.

Why? Concrete numbers make it clear. Start with the true statement $3 < 5$. Multiply both sides by $-1$ and the values become $-3$ and $-5$. But $-3$ is actually greater than $-5$ — it sits closer to zero on the number line. So the inequality has to flip to stay true: $-3 > -5$.

Example: Solve $-2x < 10$.

This needs division by $-2$. What happens to the inequality symbol when you divide by a negative? Show answerThe symbol flips. Dividing by -2 gives $x > -5$. Check: $x = 0$ gives $-2(0) = 0 < 10$ ✓. A value outside the solution like $x = -6$ gives $-2(-6) = 12 < 10$ ✗.

Example: Solve $-\frac{x}{3} \geq 4$.

Multiply both sides by -3 and flip the inequality: $$x \leq -12$$

Multi-Step Inequalities

As with equations, several steps may be needed to isolate the variable.

Example: Solve $2x - 7 < 5$.

Add 7 to both sides: $$2x < 12$$

Divide by 2: $$x < 6$$

Example: Solve $-3y + 5 \geq 14$.

Subtract 5 from both sides: $$-3y \geq 9$$

Divide by -3 and flip the inequality: $$y \leq -3$$

Example: Solve $4(m + 3) > 20$.

Distribute the 4: $$4m + 12 > 20$$

Subtract 12: $$4m > 8$$

Divide by 4: $$m > 2$$

Graphing Inequalities on a Number Line

Once you've solved an inequality, you can represent the solution visually on a number line.

For $x > 3$, you draw a number line, put an open circle at 3 (because 3 itself is not included), and shade everything to the right.

Graph of x > 3 showing open circle at 3 and shading to the right

For $x \leq -2$, you put a closed circle at -2 (because -2 is included) and shade to the left.

Graph of x ≤ -2 showing closed circle at -2 and shading to the left

Think of the open circle as saying "everything up to but not including this point" and the closed circle as "everything including this point."

Open vs. Closed Circles

The difference between $<$ and $\leq$ (or between $>$ and $\geq$) is whether the endpoint is included:

Comparison of open and closed circles on number lines

Example: Graph $x \geq -1$.

Draw a number line. Put a closed circle (or filled dot) at -1. Shade to the right, showing all numbers greater than or equal to -1.

Graph of x ≥ -1 showing closed circle at -1 and shading to the right

Example: Graph $y < 4$.

Draw a number line. Put an open circle at 4. Shade to the left, showing all numbers less than 4.

Graph of y < 4 showing open circle at 4 and shading to the left

Inequalities with Variables on Both Sides

The variable can appear on both sides of an inequality, the same way it can in an equation.

Example: Solve $5x - 3 > 2x + 9$.

Subtract $2x$ from both sides: $$3x - 3 > 9$$

Add 3: $$3x > 12$$

Divide by 3: $$x > 4$$

Example: Solve $7 - 2y \leq 3y + 17$.

Add $2y$ to both sides: $$7 \leq 5y + 17$$

Subtract 17: $$-10 \leq 5y$$

Divide by 5: $$-2 \leq y$$

This can also be written as $y \geq -2$ (same thing, just reversed).

Practice Problems

Solve and graph: $x + 4 < 9$ Show answer$x < 5$ — Graph: open circle at 5, shade left
Solve and graph: $3y \geq -12$ Show answer$y \geq -4$ — Graph: closed circle at -4, shade right
Solve: $-5m > 20$ Show answerDivide by -5, flip sign: $m < -4$
Solve: $2a - 8 \leq 6$ Show answerAdd 8: $2a \leq 14$, divide by 2: $a \leq 7$
Solve: $-4x + 3 > 15$ Show answerSubtract 3: $-4x > 12$, divide by -4 and flip: $x < -3$
Solve: $5p - 2 \geq 3p + 8$ Show answerSubtract $3p$: $2p - 2 \geq 8$, add 2: $2p \geq 10$, divide by 2: $p \geq 5$

What's Next?

The single most important habit: every time you multiply or divide both sides by a negative number, flip the inequality. That's the difference between inequalities and equations, and it's the most common place to slip. The other habits carry over from solving equations: combine like terms first when you can, and check your answer by plugging a value from the solution set back into the original inequality.

Two more details from the graphing side. The symbols $<$ and $>$ get open circles (endpoint not included); $\leq$ and $\geq$ get closed circles (endpoint included). And shading direction follows the inequality direction: $x > 5$ shades to the right, $x < 5$ shades to the left.

From here, the next step is compound inequalities, which chain two inequalities together using "and" or "or" — useful for describing ranges like "between 5 and 10."