CogniMath™ Algebra — Module 4

The Sign Switch: Why Plus Becomes Minus

What This Module Is Really About

In algebra, solving equations is not about moving numbers randomly. It is about protecting balance.

When a number crosses the equal sign, students are often told: “Just change the sign.” But the real question is — why?

The Core Idea You Must Understand

The equal sign (=) is not decoration. It is a border between two balanced sides.

When a number crosses this border, it is removed from one side and added to the other. This action automatically changes the sign.

Sign change is not a trick — it is an inverse operation.

Why This Matters

  • You stop guessing.
  • You stop memorising rules.
  • You stop making careless sign mistakes.
  • You start solving equations with confidence.

Balance Logic — Why Signs Change

Every equation works like a balance scale. Both sides must always stay equal.

When you move a number across the equal sign, you are removing it from one side and adding it to the other. This is why the sign changes.

Crossing the equal sign = applying the inverse operation

Worked Examples (Sign Switch in Action)

Example 1
x + 3 = 7
x = 7 − 3
Example 2
x − 5 = 10
x = 10 + 5
Example 3
x + 8 = 12
x = 12 − 8
Example 4
x − 2 = 9
x = 9 + 2
Example 5
x + 1 = 0
x = 0 − 1
Example 6
x − 4 = 0
x = 0 + 4

Notice that the variable does not move. Only the number crosses the equal sign — and its sign changes.

Practice — Control the Sign Switch

Set 1
  1. x + 2 = 7
  2. x − 3 = 5
  3. x + 4 = 10
Set 2
  1. x − 1 = 6
  2. x + 5 = 9
  3. x + 6 = 11
Set 3
  1. x − 4 = 3
  2. x + 7 = 12
  3. x − 2 = 8
Set 4
  1. x + 3 = 6
  2. x − 5 = 4
  3. x + 8 = 13
Set 1
  1. x + 9 = 20
  2. x − 7 = 15
  3. x + 12 = 30
Set 2
  1. x − 20 = 5
  2. x + 6 = 0
  3. x − 11 = 4
Set 3
  1. x + 14 = 25
  2. x − 9 = 6
  3. x + 18 = 40
Set 4
  1. x − 13 = 7
  2. x + 16 = 28
  3. x − 10 = 12
Set 1
  1. x + 5 = 2
  2. x − 8 = 1
  3. x + 10 = 4
Set 2
  1. x − 15 = −5
  2. x + 3 = −7
  3. x − 12 = −2
Set 3
  1. x + 9 = −1
  2. x − 20 = −10
  3. x + 4 = −6
Set 4
  1. x − 18 = −8
  2. x + 6 = −4
  3. x − 22 = −12

Visual Logic — Crossing the Equal Sign

Think of the equal sign as a border. When a number crosses this border, its sign changes to keep balance.

x + 3
➜ ➜ ➜
cross the border
⟂ = ⟂
7
+3 crosses →
becomes −3
x = 7 − 3

Notice: the variable stays in place. Only the number moves — and its sign changes.

Self-Check — Are You Doing the Sign Switch Correctly?

Before moving on, use this quick checklist to verify your thinking. If any answer is “No”, pause and re-check the step.

Check 1

Did only the number move across the equal sign?

Check 2

Did the sign change when the number crossed?

Check 3

Did the variable stay in place?

Check 4

Does the final equation still look balanced?

Common Mistakes to Avoid

Mistake 1

Changing the sign without crossing the equal sign.

Mistake 2

Moving the variable instead of the number.

Mistake 3

Forgetting that subtraction becomes addition (and vice versa).

Mistake 4

Panicking when answers become negative.

Welcome to your Module 4 Mastery Quiz — The Sign Switch

Q1) Question: Solve: x + 7 = 12

Focus on what happens to +7 when it moves across the equal sign.

Q2) Question: Solve: x − 9 = 4

Think about how subtraction is undone when solving equations.

Q3) Question: Which step is correct for solving x + 6 = 2?

Only one step keeps the equation balanced.

Q4) Question: Solve: x − 5 = −3

Watch carefully how a negative number behaves when crossing the equal sign.

Q5) Question: A student writes: x + 4 = 10 → x = 10 + 4 What is the mistake?

The student moved a number but ignored an important rule.

Q6) Question: Solve: x + 9 = 0

Zero is also a number. Treat it like any other value.

Q7) Question: Which equation gives the solution x = −4?

Test each equation by solving it mentally.

Q8) Question: Solve: x − 12 = −10

Two negative numbers are involved. Stay calm and follow balance logic.

Q9) Question: Which step keeps the equation balanced? x − 8 = 1

Only one option correctly undoes the subtraction.

Q10) Question: Why does a sign change when a number crosses the equal sign?

This question checks understanding, not calculation.

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