How to Explain Equations and Inequalities to Seventh Graders

"If I have $50 and tickets cost $8 each, plus there's a $10 service fee, how many tickets can I buy?"

This everyday question is an equation in disguise. When seventh graders learn to set up and solve equations (and inequalities), they gain a powerful tool for answering questions like this—and thousands more.

Why Equations and Inequalities Matter

Equations and inequalities model real decisions:

• Shopping: "How many items can I afford?"
• Planning: "How many hours do I need to work?"
• Cooking: "How much of each ingredient do I need?"
• Games: "What score do I need to win?"
• Science: Converting temperatures, calculating speed

These skills are foundational for:

• All higher algebra
• Functions and graphing
• Physics and chemistry
• Business and finance

Understanding Equations

What Is an Equation?

An equation states that two expressions are equal.

3x + 5 = 14
↑   ↑    ↑
Left side = Right side

The solution is the value that makes the equation true.

The Balance Model

Think of an equation as a balanced scale:

     3x + 5         =         14

    ┌───────┐              ┌───────┐
    │ 3x+5  │              │  14   │
    └───┬───┘              └───┬───┘
        │                      │
     ═══╧══════════════════════╧═══
              balanced!

Golden Rule: Whatever you do to one side, you MUST do to the other side to keep it balanced.

Solving One-Step Equations

Using Inverse Operations

Addition and subtraction are inverses:

x + 7 = 12
x + 7 - 7 = 12 - 7    (subtract 7 from both sides)
x = 5

Multiplication and division are inverses:

4x = 20
4x ÷ 4 = 20 ÷ 4      (divide both sides by 4)
x = 5

One-Step Examples

x - 9 = 15           |    x/3 = 7
x - 9 + 9 = 15 + 9   |    x/3 · 3 = 7 · 3
x = 24               |    x = 21

-5x = 35             |    x + 2.5 = 10
-5x ÷ (-5) = 35 ÷ (-5) |  x + 2.5 - 2.5 = 10 - 2.5
x = -7               |    x = 7.5

Solving Two-Step Equations

The Order Matters

Reverse the order of operations:

1. First, undo addition or subtraction
2. Then, undo multiplication or division

Think: What happened to x last? Undo that first!

Step-by-Step Process

Solve: 3x + 5 = 14

3x + 5 = 14

Step 1: Undo addition (subtract 5)
3x + 5 - 5 = 14 - 5
3x = 9

Step 2: Undo multiplication (divide by 3)
3x ÷ 3 = 9 ÷ 3
x = 3

Check: 3(3) + 5 = 9 + 5 = 14 ✓

More Examples

Solve: 2x - 7 = 15

2x - 7 = 15
2x - 7 + 7 = 15 + 7     (add 7)
2x = 22
2x ÷ 2 = 22 ÷ 2         (divide by 2)
x = 11

Check: 2(11) - 7 = 22 - 7 = 15 ✓

Solve: x/4 + 3 = 10

x/4 + 3 = 10
x/4 + 3 - 3 = 10 - 3    (subtract 3)
x/4 = 7
x/4 · 4 = 7 · 4         (multiply by 4)
x = 28

Check: 28/4 + 3 = 7 + 3 = 10 ✓

Solve: -5x + 12 = -8

-5x + 12 = -8
-5x + 12 - 12 = -8 - 12 (subtract 12)
-5x = -20
-5x ÷ (-5) = -20 ÷ (-5) (divide by -5)
x = 4

Check: -5(4) + 12 = -20 + 12 = -8 ✓

Equations with Variables on Both Sides

The Strategy

Get all variables on one side, all constants on the other.

Solve: 5x + 3 = 2x + 15

5x + 3 = 2x + 15

Step 1: Get variables on one side (subtract 2x)
5x - 2x + 3 = 2x - 2x + 15
3x + 3 = 15

Step 2: Get constants on other side (subtract 3)
3x + 3 - 3 = 15 - 3
3x = 12

Step 3: Solve for x (divide by 3)
x = 4

Check: 5(4) + 3 = 23 and 2(4) + 15 = 23 ✓

Another Example

Solve: 4x - 9 = 7x + 12

4x - 9 = 7x + 12

Subtract 4x from both sides:
-9 = 3x + 12

Subtract 12 from both sides:
-21 = 3x

Divide by 3:
-7 = x

Check: 4(-7) - 9 = -37 and 7(-7) + 12 = -37 ✓

Equations with Distributive Property

Expand, Then Solve

Solve: 3(x + 4) = 21

3(x + 4) = 21

Step 1: Distribute
3x + 12 = 21

Step 2: Subtract 12
3x = 9

Step 3: Divide by 3
x = 3

Check: 3(3 + 4) = 3(7) = 21 ✓

With Variables on Both Sides

Solve: 2(x - 3) = 4x + 8

2(x - 3) = 4x + 8

Distribute:
2x - 6 = 4x + 8

Subtract 2x:
-6 = 2x + 8

Subtract 8:
-14 = 2x

Divide by 2:
-7 = x

Check: 2(-7 - 3) = 2(-10) = -20
       4(-7) + 8 = -28 + 8 = -20 ✓

Understanding Inequalities

Inequality Symbols

<    less than
>    greater than
≤    less than or equal to
≥    greater than or equal to

Reading Inequalities

x < 5     "x is less than 5"
x > 3     "x is greater than 3"
x ≤ 7     "x is less than or equal to 7"
x ≥ -2    "x is greater than or equal to -2"

Graphing Inequalities on a Number Line

Open circle (○) for < and > (endpoint NOT included)
Closed circle (●) for ≤ and ≥ (endpoint IS included)

x < 3
←──────────○
-1  0  1  2  3  4  5

x ≥ -1
           ●──────────→
-3 -2 -1  0  1  2  3

-2 < x ≤ 4
     ○────────────●
-3 -2 -1  0  1  2  3  4  5

Solving Inequalities

Same Process as Equations (Almost!)

Solve: x + 5 > 12

x + 5 > 12
x + 5 - 5 > 12 - 5
x > 7

Solve: 3x ≤ 18

3x ≤ 18
3x ÷ 3 ≤ 18 ÷ 3
x ≤ 6

The Exception: Multiplying/Dividing by Negatives

FLIP THE SIGN when multiplying or dividing by a negative number!

Solve: -4x > 20

-4x > 20
-4x ÷ (-4) < 20 ÷ (-4)    ← Sign FLIPS!
x < -5

Why does the sign flip?

Consider: 2 < 5

Multiply both by -1:
-2 ? -5

-2 is to the RIGHT of -5 on the number line
So -2 > -5

The order reversed!

Two-Step Inequality

Solve: 2x - 7 ≥ 11

2x - 7 ≥ 11
2x - 7 + 7 ≥ 11 + 7
2x ≥ 18
2x ÷ 2 ≥ 18 ÷ 2
x ≥ 9

Graph:
           ●──────────→
6  7  8  9  10  11  12

Solve: -3x + 5 < 14

-3x + 5 < 14
-3x + 5 - 5 < 14 - 5
-3x < 9
-3x ÷ (-3) > 9 ÷ (-3)    ← Sign FLIPS!
x > -3

Graph:
     ○──────────→
-5 -4 -3 -2 -1  0  1

Writing Equations and Inequalities from Word Problems

Key Phrases

Equations (exactly equal):

• "is," "equals," "is the same as," "gives," "yields"

Inequalities:

• "at least" → ≥
• "at most" → ≤
• "more than" → >
• "fewer than," "less than" → <
• "no more than" → ≤
• "no fewer than" → ≥

Setting Up Word Problems

Example 1:
"Maria has $85. She wants to buy jeans for $35 and some shirts that cost $12 each. How many shirts can she buy?"

Let x = number of shirts

Equation: 35 + 12x = 85
(jeans + shirts = total)

12x = 50
x = 4.17

Maria can buy 4 shirts (can't buy partial shirts).

Example 2:
"A phone plan costs $25 per month plus $0.10 per text. Jake wants to spend at most $40 per month. How many texts can he send?"

Let x = number of texts

Inequality: 25 + 0.10x ≤ 40

0.10x ≤ 15
x ≤ 150

Jake can send at most 150 texts.

Checking Solutions in Context

Always ask:

• Does the answer make sense?
• Should it be a whole number?
• Is it positive (if counting objects)?

Hands-On Activities

Balance Scale Lab

Use a physical or virtual balance scale:

1. Place objects on both sides (variables = bags of unknown weight)
2. Find the weight that balances
3. Connect to algebraic solving

Inequality Card Sort

Cards with scenarios—students match to inequality symbols:

"At least 18 years old" → x ≥ 18
"Fewer than 10 items" → x < 10
"No more than $50" → x ≤ 50

Number Line Dash

On a large floor number line:

1. Teacher calls an inequality
2. Students run to stand on the correct region
3. Discuss boundary (open vs. closed circle)

Real-World Equation Hunt

Find situations that use equations:

• Cell phone plans
• Gym memberships
• Taxi fares
• Utility bills

Set up and solve the equations.

Algebra Tile Equations

Use algebra tiles to model equations:

• Green rectangles = positive x
• Yellow squares = positive 1
• Red = negatives

Physically "remove" tiles from both sides to solve.

Common Mistakes and How to Fix Them

Mistake 1: Wrong Order of Operations When Solving

Error: For 3x + 5 = 14, dividing by 3 first: x + 5 = 4.67

Fix: Always undo addition/subtraction FIRST, then multiplication/division. Think: "What happened to x last?" and undo that first.

Mistake 2: Forgetting to Flip the Inequality Sign

Error: -2x > 8, so x > -4

Fix: When dividing by -2, FLIP the sign: x < -4. Test it: try x = -5. Is -2(-5) > 8? Is 10 > 8? Yes! Try x = -3. Is -2(-3) > 8? Is 6 > 8? No. So x < -4 is correct.

Mistake 3: Sign Errors with Negatives

Error: x - 5 = -12, so x = -12 - 5 = -17

Fix: Adding 5 to both sides: x = -12 + 5 = -7. Check: -7 - 5 = -12 ✓

Mistake 4: Not Distributing to All Terms

Error: 2(x - 4) = 2x - 4

Fix: The 2 multiplies BOTH terms: 2(x - 4) = 2x - 8

Mistake 5: Confusing "Less Than" When Writing Inequalities

Error: "5 less than x" written as 5 < x

Fix: "5 less than x" means x - 5 (an expression, not an inequality). "x is less than 5" is x < 5 (an inequality). Listen for "is" to know it's an inequality.

Checking Your Work

For Equations

Substitute your answer back into the ORIGINAL equation.

Solve: 4x - 7 = 13
Solution: x = 5

Check: 4(5) - 7 = 20 - 7 = 13 ✓

For Inequalities

Test a value IN your solution region.

Solve: -2x + 5 > 11
Solution: x < -3

Test x = -4 (which is < -3):
-2(-4) + 5 = 8 + 5 = 13
Is 13 > 11? Yes! ✓

Also test a value OUTSIDE your solution to confirm it doesn't work.

Test x = 0 (which is NOT < -3):
-2(0) + 5 = 5
Is 5 > 11? No! ✓

Connecting to Other Concepts

Equations and Proportions

Proportional relationships lead to equations:

If 3 apples cost $4.50, how much do 7 apples cost?

3/4.50 = 7/x
3x = 31.50
x = $10.50

Equations and Graphs

The solution to an equation is where a line crosses a specific y-value:

y = 2x + 3

When does y = 11?
11 = 2x + 3
x = 4

The line y = 2x + 3 equals 11 when x = 4.

Inequalities and Number Sets

Inequalities describe sets of numbers:

x > 5 describes: {6, 7, 8, ...} (for integers)
or all real numbers greater than 5

To Functions

Solving equations finds specific inputs or outputs:

f(x) = 3x - 2

Find x when f(x) = 10:
10 = 3x - 2
x = 4

Practice Ideas for Home

Budget Planning

"You have $75 for school supplies. Notebooks cost $3 each and you need a $20 backpack. How many notebooks can you buy?"

20 + 3n ≤ 75
3n ≤ 55
n ≤ 18.3

At most 18 notebooks.

Temperature Conversions

F = (9/5)C + 32

What Celsius temperature is 50°F?
50 = (9/5)C + 32
18 = (9/5)C
C = 10°

Savings Goals

"You have $40 saved. You earn $8 per hour. How many hours until you have $200?"

40 + 8h = 200
8h = 160
h = 20 hours

Game Scoring

"You need at least 80 points to level up. You have 35 points and each star is worth 5 points. How many stars do you need?"

35 + 5s ≥ 80
5s ≥ 45
s ≥ 9 stars

The Bottom Line

Equations and inequalities are tools for answering questions. When students set up the mathematical relationship correctly, solving becomes a systematic process of isolating the variable.

Key takeaways:

1. Equations have one solution; inequalities have many
2. Use inverse operations to isolate the variable
3. Undo addition/subtraction before multiplication/division
4. FLIP the inequality sign when multiplying/dividing by negatives
5. Always check your answer in the original problem

When seventh graders master equations and inequalities, they can model and solve problems from budgeting to science. That's the power of algebra in action.