Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 8 chapters and video availability01 · Read and understand
What you will learn
- Explain how to recognize equations with no solution or all real solutions.
- Solve the two practice problems and explain how the assumptions affect the answers.
Before you start
Distribution and subtracting equal terms from both sides.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Classify the real solutions of 2(x + 3) = 2x + 7 and 2(x + 3) = 2x + 6.
Why this math matters
If variable terms cancel, the remaining statement tells whether an equation is impossible or true for every allowed input. Inspect the final statement after simplification before deciding whether an equation has one, none, or infinitely many solutions.
Set up the model
A useful answer starts with clear assumptions:
- x ranges over all real numbers.
- The question asks for every value satisfying each equation.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Recognize equations with no solution or all real solutions
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Classify the real solutions of 2(x + 3) = 2x + 7 and 2(x + 3) = 2x + 6.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the quantities
first equation: 2x + 6 = 2x + 7 → 6 = 7
Subtracting 2x leaves a statement that is always false.
Apply the relationship
second equation: 2x + 6 = 2x + 6 → 6 = 6
After cancellation, the equality is always true.
Check and interpret
First: no solution. Second: every real x.
Cancellation does not force x = 0; it exposes either a contradiction or an identity.
The result
First: no solution. Second: every real x.
Cancellation does not force x = 0; it exposes either a contradiction or an identity.
Common mistakes to catch
- A variable disappearing is not a reason to write x = 0.
- An identity means all allowed values, not an undefined or empty answer.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Classify 5x − 1 = 5x − 1.
Show a hint
Subtract 5x from both sides.
Reveal answer and explanation
All real numbers
The remaining statement −1 = −1 is always true.
Practice 2
Classify 4x + 3 = 4x − 2.
Show a hint
Compare the constants after cancelling 4x.
Reveal answer and explanation
No solution
The equation would require 3 = −2, which no x can change.
Take the idea with you
Inspect the final statement after simplification before deciding whether an equation has one, none, or infinitely many solutions.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Solve two linked conditions by substitution
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