LEARNER JOURNEY Solve 3(x + 2) = 3x + 6
Name the goalSimplify both sides.
Choose the next moveDistribute on the left.
Carry out the mathematics3x + 6 = 3x + 6, then 6 = 6.
Check and interpretThe statement is always true, so all real numbers solve it.
Start here
Build the concept from the beginning
Starting point
No prior equation-solving skill is assumed. Arithmetic operations, variables, equality, and every new algebra decision are explained on this page.
Learning target
Recognize when simplification produces infinitely many solutions or no solution.
Evidence of mastery
Construct the reasoning path, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- equation
- A statement that two mathematical expressions have the same value.
- solution
- A value that makes an equation true when substituted for the variable.
- coefficient
- A numerical factor multiplying a variable.
- constant
- A number without a variable.
Look first, then name the mathematics
Build the visual meaning
- True statement
- All real numbers
- Contradiction
- No solution
Read the complete visual relationship as text
- Variables cancel
- True statement
- Identity
- False statement
- No solution
Definition in plain language
If the variable cancels and a true statement remains, every allowed value is a solution. If a false statement remains, no value is a solution.
Why this matters
These outcomes are valid solution sets, not failed attempts to isolate x.
Reasoning path
See the method as a sequence
Move from left to right. Each decision prepares the next one; on a phone, follow the numbered cards from top to bottom.
Simplify both sides completely.
Collect variable terms.
Notice whether all variable terms cancel.
Classify the remaining statement as true or false.
The same method in words
- Simplify both sides completely.
- Collect variable terms.
- Notice whether all variable terms cancel.
- Classify the remaining statement as true or false.
Interactive decision model
Build the reasoning, one decision at a time
Classify 5(x + 1) = 5x + 2.
Choose this step, then check the construction.
Choose this step, then check the construction.
Choose this step, then check the construction.
The reasoning path changes as each decision is selected.
Choose every step, then check the construction.
Model before practice
Three fully explained examples
Every example identifies the goal, explains the next move, carries out the algebra, and checks or interprets the result.
Solve 4x + 1 = 4x − 5
Name the goalCollect x-terms.
Choose the next moveSubtract 4x from both sides.
Carry out the mathematics1 = −5 remains.
Check and interpretThe contradiction is never true, so there is no solution.
Solve 2(x − 4) + 8 = 2x
Name the goalSimplify the left expression.
Choose the next move2x − 8 + 8 = 2x.
Carry out the mathematics2x = 2x, then 0 = 0.
Check and interpretThis identity is true for every real x.
Practice with decreasing support
Guided practice
Practice immediately after the model
Predict first. Then use the visual relationship and reasoned feedback to check two decisions.
1. Solve 3(x + 2) = 3x + 6.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve 4x + 5 = 4x − 2.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. Solve 2(x − 3) = 2x − 6.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Independent practice
Six problems for repetition
Solve all six. Use the specific feedback to revise a method or sign error.
1. Solve 5x + 1 = 5x + 1.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve 7x − 4 = 7x + 9.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. What result signals an identity?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. What result signals no solution?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Solve 6(x + 1) − 6x = 6.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
6. Solve 3(x + 4) = 3x + 10.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. Solve 8x − 2(4x − 3) = 6.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve 9x + 7 = 9x + 7.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Solve 2x + 5 = 2x + 4.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. If an equation simplifies to 0 = 0, how many real solutions does it have?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. If an equation simplifies to 0 = 5, how many solutions does it have?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Compare an equation that has x = 0 as its only solution with an equation that has no solution.
Guided self-check
Judge the locked first attempt against every criterion. Each judgment gives revision guidance.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Model reasoning
The equation 3x = 0 has the single solution x = 0 because substitution gives 0 = 0. The equation 3x + 1 = 3x + 4 has no solution because subtracting 3x leaves 1 = 4. Zero is a valid value; no solution means no value works.
Write a first attempt before opening the self-check.
The record reveals exact answers and model reasoning only after the corresponding attempt or self-check has occurred.