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\( < \)
\( > \)

One-Step
Inequalities

Learn to solve inequalities like \( x + 5 > 12 \) and \( 3x \leq 18 \), then graph solutions on a number line.

Pre-Algebra

What Is an Inequality?

An inequality is a mathematical statement that compares two expressions using a symbol other than \( = \). Instead of one exact answer, an inequality describes a range of values that make the statement true.

\( < \) Less Than
\( x < 5 \) means \( x \) is any number below 5
\( > \) Greater Than
\( x > 5 \) means \( x \) is any number above 5
\( \leq \) Less Than or Equal
\( x \leq 5 \) includes 5 itself
\( \geq \) Greater Than or Equal
\( x \geq 5 \) includes 5 itself

Graphing on a Number Line

We show inequality solutions on a number line using circles and arrows. The circle sits on the boundary value, and the arrow shows which direction the solutions go.

0 3 6 -3
\( x > 3 \): open circle, arrow right
0 3 6 -3
\( x \leq 3 \): closed circle, arrow left

Equations vs Inequalities

You already know how to solve equations. Inequalities work almost the same way, with one critical extra rule.

Equation
\( x + 5 = 12 \) has one solution: \( x = 7 \)
Inequality
\( x + 5 > 12 \) has infinitely many solutions: \( x > 7 \)

The Addition/Subtraction Rule

You can add or subtract the same number on both sides of an inequality without changing the direction of the symbol.

If \( a > b \), then
\( a + c > b + c \)
\( a - c > b - c \)

The Multiply/Divide Rule

You can multiply or divide both sides by a positive number and the symbol stays the same. But if you multiply or divide by a negative number, you must flip the inequality symbol.

Positive number
\( 2x > 10 \)
\( x > 5 \)
 Symbol stays ✓ Negative number
\( -2x > 10 \)
\( x < -5 \)
 Symbol FLIPS !

Strict vs Non-Strict Inequalities

Strict Non-Strict
Symbols \( < \) and \( > \) \( \leq \) and \( \geq \)
Boundary included? No Yes
Number line circle Open \( \circ \) Closed \( \bullet \)
Example \( x > 4 \): 4 is NOT a solution \( x \geq 4 \): 4 IS a solution

How to Solve Step by Step

1
Identify the operation being done to the variable (addition, subtraction, multiplication, or division).
2
Use the inverse operation on both sides to isolate the variable.
3
Check: did you multiply or divide by a negative? If yes, flip the inequality symbol.
4
Graph the solution on a number line with the correct circle (open or closed) and arrow direction.

Example 1: \( x + 5 > 12 \)

1
Operation: \( 5 \) is being added to \( x \).
2
Inverse: subtract \( 5 \) from both sides: \( x + 5 - 5 > 12 - 5 \), so \( x > 7 \).
3
Flip check: No negative multiplication or division. Symbol stays \( > \).
4 5 6 7 8 open at 7, arrow right

Example 2: \( 3x \leq 18 \)

1
Operation: \( x \) is being multiplied by \( 3 \).
2
Inverse: divide both sides by \( 3 \): \( \frac{3x}{3} \leq \frac{18}{3} \), so \( x \leq 6 \).
3
Flip check: Divided by positive 3. Symbol stays \( \leq \).
3 4 5 6 7 closed at 6, arrow left

Example 3: \( -4x > 20 \)

1
Operation: \( x \) is being multiplied by \( -4 \).
2
Inverse: divide both sides by \( -4 \): \( \frac{-4x}{-4} \;\;?\;\; \frac{20}{-4} \).
3
Flip! Divided by a negative, so \( > \) becomes \( < \). Result: \( x < -5 \).
-8 -7 -6 -5 -4 open at -5, arrow left

Why Does the Symbol Flip?

Think about the number line. We know \( 3 > 1 \). Now multiply both sides by \( -1 \):

-3 -2 -1 0 1 3 1 3
\( 3 > 1 \)
 -1 -3
\( -3 < -1 \)

Example 4: \( x - 9 \geq -2 \)

1
Operation: \( 9 \) is being subtracted from \( x \).
2
Inverse: add \( 9 \) to both sides: \( x - 9 + 9 \geq -2 + 9 \), so \( x \geq 7 \).
3
Flip check: No negative multiplication or division. Symbol stays \( \geq \).
4 5 6 7 8 closed at 7, arrow right

Example 5: \( \frac{x}{-3} \leq 4 \)

1
Operation: \( x \) is being divided by \( -3 \).
2
Inverse: multiply both sides by \( -3 \).
3
Flip! Multiplying by a negative, so \( \leq \) becomes \( \geq \). Result: \( x \geq -12 \).
-15 -12 -9 -6 closed at -12, arrow right

Common Mistakes

✕ Forgetting to Flip
When dividing or multiplying by a negative, students forget to reverse the inequality symbol. Always ask: "Did I use a negative?"
✕ Wrong Circle Type
Using a closed circle for \( < \) or \( > \), or an open circle for \( \leq \) or \( \geq \). Remember: line under the symbol means filled.
✕ Arrow in Wrong Direction
Drawing the arrow to the right for \( x < 5 \). The arrow should always point toward the solutions: left for "less than", right for "greater than".

Summary Reference Table

Operation Symbol Change? Example
Add to both sides No change \( x - 3 > 5 \;\Rightarrow\; x > 8 \)
Subtract from both sides No change \( x + 7 \leq 10 \;\Rightarrow\; x \leq 3 \)
Multiply/divide by positive No change \( 4x < 20 \;\Rightarrow\; x < 5 \)
Multiply/divide by negative FLIP ! \( -2x \geq 6 \;\Rightarrow\; x \leq -3 \)

Question 1 of 5

Solve: \( x + 8 > 15 \)

Question 2 of 5

Solve: \( -5x \leq 30 \)

Question 3 of 5

Which number line correctly represents \( x < 3 \)?

Question 4 of 5

Solve: \( \frac{x}{-2} \geq 7 \)

Question 5 of 5

A student solves \( x - 4 \leq 10 \) and writes \( x \leq 6 \). What went wrong?