Learn with Amar
Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Track sign changes while excluding a denominator zero.
- Justify the conclusion "Solution: (−∞,−1)∪[2,∞)" using the stated assumptions.
Before you start
Factoring and inequalities.
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
Solve (x−2)/(x+1)≥0.
Why this math matters
Track sign changes while excluding a denominator zero. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The inequality is over real inputs.
- The denominator must be nonzero.
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.
Solve a rational inequality by intervals
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve (x−2)/(x+1)≥0.
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.
Build the model
Critical values are x=−1 and x=2
The denominator changes sign at −1; the numerator vanishes at two.
Work through the mathematics
Signs are positive on (−∞,−1), negative on (−1,2), positive on (2,∞)
Test one point in each interval because signs stay constant between critical values.
Check the conclusion
Solution: (−∞,−1)∪[2,∞)
Include the zero numerator at two but exclude the undefined input −1.
The result
Solution: (−∞,−1)∪[2,∞)
Include the zero numerator at two but exclude the undefined input −1.
Common mistakes to catch
- A denominator zero is never included.
- Squaring or cross-multiplying can lose sign information.
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
Does x=−2 satisfy the inequality?
Show a hint
Substitute directly.
Reveal answer and explanation
Yes
(−4)/(−1)=4≥0.
Practice 2
Why not multiply by x+1 without cases?
Show a hint
Its sign is unknown.
Reveal answer and explanation
It may reverse the inequality
Multiplying by a negative quantity changes the direction, and zero is not allowed.
Take the idea with you
Use interval sign reasoning to identify when a quotient-based model has a nonnegative output.
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: Reject a root introduced by squaring
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