Solve And Graph Inequalities

Solve inequalities faster by thinking in ranges on a number line, then translate that picture into clean notation. Learn the one rule that changes algebra steps, handle AND versus OR compounds, and treat absolute value as distance so graphs and answers match.

An inequality is describing a whole set of numbers that make a statement true. Once you picture that set as a shaded part of a number line, the symbols, the algebra steps, and the graph all start agreeing with each other.

Inequalities are ranges, not points

The symbols tell you which side of a boundary is allowed.

• Inequality symbols < and > mean strictly less or strictly greater, so the boundary value is excluded.
• ≤ and ≥ include the boundary value.

On a number line, that boundary is shown with a circle at the boundary number. An open circle means the endpoint is not included. A closed circle means it is included. The shading shows all values that work.

Use this number line view to connect each symbol to its graph.

A useful mental check is to read the inequality like a rule for picking numbers. For $x>2$, you could pick $3$, $100$, or $2.1$. You cannot pick $2$.

Range mindset If the solution feels like it should be many numbers, but you wrote one number, you probably solved it like an equation instead of an inequality.

Solving linear inequalities without losing the sign

To solve most linear inequalities, do the same algebra you do for equations. Combine like terms, move terms across, then isolate the variable. The one extra rule is the reason people get stuck.

The rule that flips the symbol

If you multiply or divide both sides by a negative number, the inequality sign reverses.
Example: from $x<5$ you get $-x>-5$ after multiplying by $-1$.

Why. Multiplying by a negative reflects the number line across zero. Order reverses, so the symbol must reverse to keep the statement true.

Watch a step-by-step solve where the flip is highlighted exactly at the moment it matters.

Quick examples

If $x+4\le 10$, subtract $4$ to get $x\le 6$. No flip, because you subtracted.

If $-2x>8$, divide by $-2$ and flip the sign to get $x<-4$.

The goal is to keep the same set of solutions while you rewrite the inequality into a simpler form.

Graphing and writing solutions cleanly

A complete answer often has two parts. The graph on a number line and the notation that matches it. Here is the clean translation.

Interval notation

Interval notation uses parentheses for excluded endpoints and brackets for included endpoints.

• $x<3$ becomes $(-\infty,3)$
• $x\ge -1$ becomes $[-1,\infty)$
• $-2<x\le 5$ becomes $(-2,5]$

Infinity is never included, so it always uses parentheses.

Set-builder notation

Set-builder describes the same set using a condition.

• $\{x\mid x<3\}$
• $\{x\mid x\ge -1\}$
• $\{x\mid -2<x\le 5\}$

If you are unsure your graph and notation match, pick a test value from the shaded region and plug it into the original inequality. It should be true. Pick one from outside the shading. It should be false.

Use the mapping tool to go from a shaded graph to the matching interval and set-builder form, then confirm with a test point.

Compound inequalities with AND and OR

A compound inequality is either an overlap problem or a split problem. The word you see tells you which.

AND means overlap

Example: $3<x\le 7$ means $x$ is greater than $3$ AND at the same time less than or equal to $7$. The solution is the overlap, one continuous segment.

You can often solve these by doing the same operation to all three parts at once.
If $1\le 2x+3<9$, subtract $3$ to get $-2\le 2x<6$, then divide by $2$ to get $-1\le x<3$.

OR means either side works

Example: $x<3$ OR $x>7$ means numbers on either side of the middle gap. The solution is two rays, not one segment.

Compare the two pictures side by side so your brain stops mixing them up.

One picture rule AND gives one shaded chunk. OR usually gives two separate shaded pieces. If your graph shape does not match that, re-check the logic word.

Absolute-value inequalities as distance

Absolute value measures distance from zero, and more generally $|x-a|$ measures distance from $a$ on the number line. That makes the patterns predictable.

When it is less than

$|x-a|<b$ means the distance from $a$ is smaller than $b$. You are inside a window around $a$.

So it becomes a double inequality:
$a-b<x<a+b$

Example: $|x-2|<5$ becomes $-3<x<7$.

When it is greater than or equal to

$|x-a|\ge b$ means the distance from $a$ is at least $b$. You are outside the window, on either side.

So it becomes an OR:
$x\le a-b$ OR $x\ge a+b$

Example: $|x-2|\ge 5$ becomes $x\le -3$ OR $x\ge 7$.

Use the distance view to see why one case creates an interval and the other creates two rays.

Common pitfalls and fast self-checks

Most mistakes come from answers that do not match the picture of what should be true. These quick checks catch them.

• If you multiplied or divided by a negative and did not flip the sign, assume it is wrong until proven otherwise.
• If the problem says at most or at least, the boundary is included. That means ≤ or ≥ and a closed circle.
• If you solved an AND compound and got two separate pieces, you probably treated it like OR.
• If you solved an OR compound and got one segment, you probably treated it like AND.
• If a test value from your shaded region makes the original inequality false, your graph is lying.

Toggle common wrong and right work to train your eyes to spot the exact moment the mistake happens.

Where inequalities show up in real decisions

Inequalities are how math talks about constraints. A budget limit is an inequality. A minimum score to pass is an inequality. A recipe that must use at least $2$ cups of flour is an inequality. In each case, the solution set is not just an answer. It is a set of allowed choices.

Translate words into symbols carefully.

• at least means ≥
• at most means ≤
• more than means >
• less than means <

Once the inequality is written, the graph is a decision map. Shaded means allowed. Unshaded means not allowed. That is why checking with test values feels so natural. You are literally testing a possible decision against the rule.

Next time you solve one, do one extra step. After you finish the algebra, say out loud what the solution means on the number line. If your sentence and your graph disagree, fix the graph before you move on.