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 9 chapters and video availability01 · Read and understand
What you will learn
- Interpret identities and contradictions after simplification.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Signed arithmetic, fraction and decimal operations, one-step equations, and introductory coordinate graphs.
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 3(x+2)=3x+6 and 3(x+2)=3x+8 over the real numbers.
Why this math matters
Recognize whether comparing two rules identifies a crossing, complete agreement, or incompatibility. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use the real-number system and the restrictions or data model stated in the question unless another domain is specified.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
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.
An equation can have all, one, or no solutions
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Classify 3(x+2)=3x+6 and 3(x+2)=3x+8 over the real numbers.
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 conditions
First equation becomes 3x+6=3x+6
Both sides are identical for every real input.
Develop the calculation
Second becomes 6=8 after subtracting 3x
This remaining statement is false regardless of x.
Check and interpret
First: all real numbers; second: no solution
Canceling the variable does not automatically produce x=0.
The result
First: all real numbers; second: no solution
Canceling the variable does not automatically produce x=0.
Common mistakes to catch
- A contradiction has no solution, not zero as a solution.
- An identity allows every input in the original domain, which may include restrictions.
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 2x+5=11.
Show a hint
Isolate x.
Reveal answer and explanation
Exactly one solution, x=3
The nonzero x coefficient remains, yielding one value.
Practice 2
What does 0x=0 tell you about x?
Show a hint
Zero times any real number is zero.
Reveal answer and explanation
Every real x works
The statement places no restriction on x within the chosen real domain.
Take the idea with you
Recognize whether comparing two rules identifies a crossing, complete agreement, or incompatibility.
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: Clear fractions in a linear inequality
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.