Do I flip the inequality symbol whenever I see a negative number?

No. You flip only when you multiply or divide *both sides* by a negative number. Adding or subtracting a negative never flips the symbol. In x - 9 > -4 you add 9 to both sides and get x > 5 with no flip, even though a negative number appeared.

How do I know whether to shade left or right?

Pick any number from the side you plan to shade and substitute it into the original inequality. If the statement comes out true, that side is correct. Never rely on the direction the symbol points unless the variable is alone on the left side.

Why does my answer have infinitely many solutions instead of one?

An equation asks which value makes two expressions exactly equal, so it usually has one answer. An inequality asks which values make one side larger or smaller, and typically a whole range of numbers does that. That's why the answer is written as an inequality and drawn as a shaded region.

What is the difference between an open and closed circle again?

Open (unfilled) circles go with , meaning the boundary number itself does not work. Closed (filled) circles go with ≤ and ≥, meaning the boundary number does work. Test the boundary value in the original inequality — if it makes a true statement, fill the circle in.

M7MATH-6.4

Solving & Graphing Inequalities

Learn to solve one- and two-step inequalities like px + q > r, flip the symbol when multiplying or dividing by a negative, and graph solutions on a number line.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Solving & Graphing Inequalities, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Solving & Graphing Inequalities free

What this lesson covers

You already know how to solve equations like

3x + 5 = 20

. Inequalities work almost the same way — but instead of one answer, you get a whole range of answers. The solution to

3x + 5 > 20

isn't a single number; it's every number bigger than 5. That means your final answer is a set, and the clearest way to show a set is to draw it on a number line.

This lesson covers two things that trip students up. First, the one rule that makes inequalities different from equations: multiplying or dividing both sides by a negative number reverses the direction of the inequality symbol. Second, graphing — knowing when to use an open circle versus a closed circle, and which direction to shade. Get those two right and inequalities become just as routine as the equations you already solve.

What an Inequality Solution Actually Means

An equation like

x = 4

has one solution. An inequality like

x > 4

has infinitely many: 4.1, 5, 7, 100, 3,000. The solution set is every value of the variable that makes the statement true.

Because you can't list infinitely many numbers, you show them two ways: with a simplified inequality such as

x > 4

, and with a number-line graph.

The four symbols and how they are read:

Symbol	Read as	Includes the endpoint?	Circle on graph
$>$	greater than	no	open
$<$	less than	no	open
$\geq$	greater than or equal to	yes	closed (filled)
$\leq$	less than or equal to	yes	closed (filled)

To graph, put your circle at the boundary number, then shade toward every number that works. For

x > 4

, shade right. For

x \leq -2

, shade left from a filled circle at

-2

.

A quick way to be sure you shaded the correct direction: pick a test number from the shaded side and plug it into the original inequality. If it makes a true statement, your shading is right. If the graph of

x > 4

is shaded left, testing

x = 0

gives

0 > 4

, which is false — so the shading must go the other way.

Students often memorize "the arrow points the same way as the symbol." That works only when the variable is written on the left. In

7 < x

, the symbol points left but the solution is all numbers greater than 7. Rewrite it as

x > 7

first, then graph.

Solving One- and Two-Step Inequalities

Solve an inequality exactly the way you solve an equation: undo the operations in reverse order to isolate the variable. Add or subtract first, then multiply or divide.

For

2x + 9 > 15

2x + 9 - 9 > 15 - 9

2x > 6

\frac{2x}{2} > \frac{6}{2}

x > 3

Adding or subtracting the same amount on both sides never changes the direction of the symbol. If 10 is greater than 4, then 10 minus 100 is still greater than 4 minus 100 — both sides shifted by the same amount, so their order is unchanged. Multiplying or dividing by a positive number is also safe.

A good habit is to check with one test value, not just re-read your steps. For

x > 3

, try

x = 5

2(5) + 9 = 19

, and

19 > 15

is true. Then try

x = 3

itself:

2(3) + 9 = 15

, and

15 > 15

is false, which confirms 3 is the boundary but not part of the solution — an open circle.

Where students go wrong most often here is treating the inequality symbol as an equals sign in the final answer and writing

x = 3

. The symbol carries meaning through every line of work; copy it down each time, and only change it for the one reason described in the next section.

The Reversal Rule: Multiplying or Dividing by a Negative

Start with a true statement:

6 > 2

. Multiply both sides by

-1

and keep the symbol:

-6 > -2

. That's false, because

-6

sits to the left of

-2

on the number line. To keep the statement true, the symbol must flip:

-6 < -2

.

That is the rule. When you multiply or divide both sides of an inequality by a negative number, reverse the inequality symbol. Multiplying by a negative reflects both numbers across zero, and reflecting reverses their order.

Step	Does the symbol flip?
Add 7 to both sides	no
Subtract 12 from both sides	no
Multiply both sides by 5	no
Divide both sides by $-3$	yes
Multiply both sides by $-\frac{1}{2}$	yes

Example: solve

-4x \leq 20

. Divide both sides by

-4

and flip:

x \geq -5

. Test

x = 0

-4(0) = 0

and

0 \leq 20

is true, and 0 is indeed greater than

-5

. Good.

Two common errors are worth naming. The first is flipping the symbol just because a negative number shows up somewhere. In

x - 8 > -3

, you add 8 to both sides — no multiplying or dividing by a negative happened, so the answer is

x > 5

with no flip. The second is subtracting a negative and flipping out of habit. Only the coefficient step matters.

Writing and Graphing the Final Answer

A complete answer to an inequality problem usually has three parts: the algebraic steps, the simplified inequality, and the graph.

Work through

-3x + 7 \geq 22

fully. Subtract 7 from both sides:

-3x \geq 15

. Divide both sides by

-3

and reverse the symbol:

x \leq -5

. To graph, place a closed circle at

-5

because

\leq

includes the endpoint, then shade left with an arrow, since every number less than

-5

also works.

When an inequality comes from a word problem, the graph and the answer often need a sentence of interpretation. If

x

is the number of hours worked, then

x \leq -5

would be impossible, which is a signal to recheck the setup. Real quantities such as hours, tickets, and people can't be negative, and sometimes they must be whole numbers — you might report "at most 5 tickets" even though the algebra allows 5.4.

Watch for the words that set up the symbol. "At least" means

\geq

, "at most" means

\leq

, "more than" means

>

, "fewer than" and "under" mean

<

, and "no more than" means

\leq

. Mixing up "at least" with "less than" changes both the symbol and the shading direction, so slow down on that translation step before doing any algebra.

Checking Your Work Like a Pro

Every inequality answer can be verified in about twenty seconds, which is why careless mistakes here are so fixable.

Use a three-number check. First, plug in the boundary value. It should make the two sides exactly equal — that confirms you did the arithmetic right. Second, plug in a number from your shaded region. It should make the original inequality true. Third, plug in a number from the unshaded side. It should make the original inequality false.

Run that on

5 - 2x < 11

. Solving: subtract 5 to get

-2x < 6

, then divide by

-2

and flip to get

x > -3

Test value	Substitute into $5 - 2x < 11$	Result	Expected
$x = -3$	$5 + 6 = 11$	$11 < 11$ false, sides equal	boundary, open circle
$x = 0$	$5 - 0 = 5$	$5 < 11$ true	in solution
$x = -10$	$5 + 20 = 25$	$25 < 11$ false	outside solution

All three behave as predicted, so

x > -3

with an open circle at

-3

and shading right is correct.

The single most valuable part of this check is the third test. If you forgot to flip a symbol, your answer would have been

x < -3

, and testing

x = -10

would have produced a false statement inside your own shaded region — an immediate red flag. Substituting into the original inequality matters; substituting into a middle line repeats any error you already made.

Key terms

Inequality.: A mathematical statement comparing two expressions with $<$ , $>$ , $\leq$ , or $\geq$ instead of an equals sign.
Solution set.: All values of the variable that make an inequality true; usually infinitely many, shown as a simplified inequality and a number-line graph.
Boundary value.: The number where the shading starts on a graph; it makes the two sides of the inequality exactly equal.
Open circle.: An unfilled circle used for $<$ or $>$ , showing the boundary value itself is not part of the solution.
Closed circle.: A filled circle used for $\leq$ or $\geq$ , showing the boundary value is included in the solution.
Reversal rule.: Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol.
Inverse operations.: Operations that undo each other, such as addition and subtraction or multiplication and division, used to isolate the variable.
Coefficient.: The number multiplied by the variable; its sign determines whether the final division step flips the symbol.

Worked example

Solve

-6x + 4 \leq 34

, graph the solution set on a number line, and verify your answer with a test value.

Start by undoing the addition. Subtract 4 from both sides:

-6x + 4 - 4 \leq 34 - 4

-6x \leq 30

Subtraction never changes the direction of the symbol, so

\leq

stays as it is.

Now undo the multiplication by

-6

. Divide both sides by

-6

. Because

-6

is negative, reverse the symbol:

\frac{-6x}{-6} \geq \frac{30}{-6}

x \geq -5

Graph it: place a closed circle at

-5

, since

\geq

includes the endpoint, then shade to the right toward larger numbers and draw an arrow on the end to show the shading continues forever.

Check the boundary. Substitute

x = -5

into the original:

-6(-5) + 4 = 30 + 4 = 34

, and

34 \leq 34

is true, so

-5

belongs in the solution — the closed circle is correct.

Check inside the shaded region. Substitute

x = 0

-6(0) + 4 = 4

, and

4 \leq 34

is true.

Check outside. Substitute

x = -10

-6(-10) + 4 = 60 + 4 = 64

, and

64 \leq 34

is false, exactly as expected for a value not in the solution.

All three tests agree, so the answer is

x \geq -5

Practice questions

Which inequality is the correct solution to

8 - 5x > 43

$x > -7$
$x < -7$
$x > 7$
$x < 7$

Answer: $x < -7$

Subtract 8 from both sides to get

-5x > 35

; subtraction does not affect the symbol. Now divide both sides by

-5

, which is negative, so the symbol reverses:

x < -7

. Check with

x = -10

8 - 5(-10) = 8 + 50 = 58

, and

58 > 43

is true, so

-10

should be in the solution — and it is, since

-10 < -7

. The most common wrong answer is

x > -7

, which comes from dividing by the negative without reversing the symbol.

Marisol is graphing the solution to

\frac{x}{3} - 2 \leq 1

. Solve the inequality, describe the graph exactly (circle type, position, and shading direction), and explain why the endpoint is or is not included.

Answer: $x \leq 9$ ; a closed circle at 9 with shading to the left and an arrow, because substituting $x = 9$ makes both sides equal 1 and $1 \leq 1$ is true.

Add 2 to both sides:

\frac{x}{3} \leq 3

. Then multiply both sides by 3. Since 3 is positive, the symbol does not flip:

x \leq 9

. The symbol

\leq

means "or equal to," so the boundary value 9 is part of the solution and gets a filled-in circle. Substituting

x = 9

gives

\frac{9}{3} - 2 = 1

, and

1 \leq 1

is true, confirming inclusion. Shade left because smaller values also work: testing

x = 0

gives

-2 \leq 1

, which is true.

A student solves

-2x + 6 < 14

and writes: "

-2x < 8

, so

x < -4

." Identify the error, give the correct solution, and show one test value that proves the student's answer is wrong.

Answer: The student divided by $-2$ without reversing the symbol; the correct solution is $x > -4$ . Testing $x = -10$ in the original gives $26 < 14$ , which is false even though $-10$ fits the student's answer.

The first step, subtracting 6, is correct and gives

-2x < 8

. The error is the second step: dividing both sides by the negative number

-2

requires reversing

<

>

, producing

x > -4

. The test value exposes the mistake because

-10

satisfies

x < -4

but makes the original inequality false. This is why substituting a value from your shaded region back into the original inequality is worth the few seconds it takes.

FAQ

Do I flip the inequality symbol whenever I see a negative number?: No. You flip only when you multiply or divide both sides by a negative number. Adding or subtracting a negative never flips the symbol. In $x - 9 > -4$ you add 9 to both sides and get $x > 5$ with no flip, even though a negative number appeared.
How do I know whether to shade left or right?: Pick any number from the side you plan to shade and substitute it into the original inequality. If the statement comes out true, that side is correct. Never rely on the direction the symbol points unless the variable is alone on the left side.
Why does my answer have infinitely many solutions instead of one?: An equation asks which value makes two expressions exactly equal, so it usually has one answer. An inequality asks which values make one side larger or smaller, and typically a whole range of numbers does that. That's why the answer is written as an inequality and drawn as a shaded region.
What is the difference between an open and closed circle again?: Open (unfilled) circles go with $<$ and $>$ , meaning the boundary number itself does not work. Closed (filled) circles go with $\leq$ and $\geq$ , meaning the boundary number does work. Test the boundary value in the original inequality — if it makes a true statement, fill the circle in.

Learn this with a teacher, not a page

The Crimsora tutor teaches Solving & Graphing Inequalities live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.

Try a live session free