One-Variable Inequalities - Seventh Grade

Symbols, Solutions, Graphing, Solving & Word Problems

1. Understanding Inequalities

What is an Inequality?

An inequality is a mathematical statement that

compares two values showing that one is

greater than, less than, or not equal to the other

Inequality Symbols

Symbol	Meaning	Example
<	Less than	x < 5
>	Greater than	x > 3
≤	Less than or equal to	x ≤ 7
≥	Greater than or equal to	x ≥ 2

Difference: Equations vs Inequalities

Equation: x + 3 = 7 (ONE solution: x = 4)

Inequality: x + 3 < 7 (MANY solutions: x < 4)

Solutions: x can be 3, 2, 1, 0, -1, -2, etc.

2. Solutions to Inequalities

What is a Solution?

A solution to an inequality is any value that

makes the inequality TRUE

Testing Solutions

Is x = 6 a solution to x > 4?

Substitute 6 for x:

6 > 4 ✓

TRUE, so x = 6 IS a solution

Is x = 2 a solution to x > 4?

Substitute 2 for x:

2 > 4 ✗

FALSE, so x = 2 is NOT a solution

Infinite Solutions

Inequalities usually have INFINITELY MANY solutions!

Example: x < 5 includes 4, 3, 2, 1, 0, -1, -2, -3, ...

3. Graphing Inequalities on Number Lines

Key Rules for Graphing

Open Circle (○): Use for < or >

The number is NOT included in the solution

Closed Circle (●): Use for ≤ or ≥

The number IS included in the solution

Arrow Direction:

• Left arrow (←) for < or ≤

• Right arrow (→) for > or ≥

Graphing Examples

Example 1: Graph x > 3

• Use open circle at 3

• Arrow pointing RIGHT

————○═══════▶
3 4 5

Example 2: Graph x ≤ 2

• Use closed circle at 2

• Arrow pointing LEFT

◀═══════●————
0 1 2 3

Example 3: Graph x ≥ -1

• Use closed circle at -1

• Arrow pointing RIGHT

————●═══════▶
-1 0 1 2

Writing Inequalities from Number Lines

Step 1: Look at the circle (open or closed)

Step 2: Look at the arrow direction

Step 3: Write the inequality

4. Solving One-Step Inequalities

Rules for Solving Inequalities

Rule 1: Add or subtract the same number from both sides

→ Inequality stays the SAME

Rule 2: Multiply or divide both sides by a POSITIVE number

→ Inequality stays the SAME

Rule 3: Multiply or divide both sides by a NEGATIVE number

→ FLIP the inequality symbol! ⚠️

⚠️ The Special Rule (Most Important!)

When multiplying or dividing by

a NEGATIVE number,

FLIP THE INEQUALITY!

< becomes >

> becomes <

Type 1: Addition/Subtraction

Solve: x + 5 < 12

Subtract 5 from both sides:

x + 5 - 5 < 12 - 5

x < 7

Check: Try x = 6: 6 + 5 = 11 < 12 ✓

Solution: x < 7

Type 2: Multiplication (Positive)

Solve: 3x ≥ 15

Divide both sides by 3 (positive):

3x ÷ 3 ≥ 15 ÷ 3

x ≥ 5

Inequality stays the SAME ✓

Solution: x ≥ 5

Type 3: Multiplication (NEGATIVE) ⚠️

Solve: -2x > 8

Divide both sides by -2 (negative):

-2x ÷ (-2) ? 8 ÷ (-2)

⚠️ FLIP THE INEQUALITY! > becomes <

x < -4

Check: Try x = -5: -2(-5) = 10 > 8 ✓

Solution: x < -4

Type 4: Division

Solve: x/4 ≤ 3

Multiply both sides by 4 (positive):

x ≤ 12

Solution: x ≤ 12

5. Solving Two-Step Inequalities

Steps to Solve

Step 1: Add or subtract to isolate the term with x

Step 2: Multiply or divide to solve for x

Step 3: Remember to flip if dividing by negative!

Example 1: Basic Two-Step

Solve: 2x + 3 < 11

Step 1: Subtract 3 from both sides

2x + 3 - 3 < 11 - 3

2x < 8

Step 2: Divide by 2 (positive)

x < 4

Graph:

◀═══════○————
2 3 4 5

Solution: x < 4

Example 2: With Negative Coefficient ⚠️

Solve: -3x + 7 ≥ 16

Step 1: Subtract 7 from both sides

-3x ≥ 9

Step 2: Divide by -3 (negative)

⚠️ FLIP THE INEQUALITY! ≥ becomes ≤

x ≤ -3

Solution: x ≤ -3

Example 3: With Fractions

Solve: x/2 - 4 > 1

Add 4 to both sides:

x/2 > 5

Multiply by 2:

x > 10

Solution: x > 10

6. Inequality Word Problems

Key Translation Words

"at least" → ≥

"at most" → ≤

"more than" → >

"less than" → <

"no more than" → ≤

"no less than" → ≥

"minimum" → ≥

"maximum" → ≤

Example 1: One-Step Word Problem

Problem: Sarah wants to buy apples that cost $3 each. She has at most $15. How many apples can she buy?

Step 1: Let x = number of apples

"at most $15" means ≤ 15

Step 2: Write inequality

3x ≤ 15

Step 3: Solve

x ≤ 5

Answer: Sarah can buy at most 5 apples

Example 2: Two-Step Word Problem

Problem: A gym charges a $20 membership fee plus $5 per visit. Maria has $65 to spend. How many times can she visit?

Let x = number of visits

Inequality: 20 + 5x ≤ 65

Solve:

5x ≤ 45

x ≤ 9

Answer: Maria can visit at most 9 times

Quick Reference: Inequality Rules

Operation	Flip Inequality?
Add or subtract same number	NO
Multiply/divide by POSITIVE number	NO
Multiply/divide by NEGATIVE number	YES ⚠️

Graphing Guide

Symbol	Circle Type	Arrow Direction
<	Open ○	Left ←
>	Open ○	Right →
≤	Closed ●	Left ←
≥	Closed ●	Right →

💡 Important Tips to Remember

✓ Most important rule: FLIP the inequality when multiplying/dividing by negative!

✓ Open circle (○): Use for < or > (number NOT included)

✓ Closed circle (●): Use for ≤ or ≥ (number IS included)

✓ Arrow left (←): Use for < or ≤

✓ Arrow right (→): Use for > or ≥

✓ Inequalities have many solutions (usually infinite)

✓ Test your solution: Pick a number and substitute

✓ "At least" means ≥, "at most" means ≤

✓ Solve inequalities just like equations (but watch for negatives!)

✓ Always check: Does your answer make sense?

🧠 Memory Tricks & Strategies

The Negative Rule:

"When you multiply or divide by negative, flip the sign - don't be tentative!"

Circle Types:

"Open circle leaves them out, closed circle lets them in - remember this without a doubt!"

Less Than Symbol:

"The alligator eats the bigger number - < points to the smaller one, remember!"

At Least vs At Most:

"At LEAST means ≥ (greater), At MOST means ≤ (lesser) - know this to be the best tester!"