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
- Use a boundary line and a test point to identify a solution region.
- 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
Describe the graph of y ≤ 2x+1.
Why this math matters
Interpret a two-variable resource constraint as a region rather than a single line. 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.
Shade a half-plane from a linear inequality
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Describe the graph of y ≤ 2x+1.
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
Boundary: y=2x+1
Equality separates the plane into two candidate regions.
Develop the calculation
At (0,0): 0≤1 is true
A point off the boundary identifies the included side.
Check and interpret
Shade on and below a solid boundary line
The ≤ sign includes all boundary points as well as the lower half-plane.
The result
Shade on and below a solid boundary line
The ≤ sign includes all boundary points as well as the lower half-plane.
Common mistakes to catch
- A test point on the boundary cannot choose a side.
- Use a solid boundary exactly when equality is included.
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
For y>−x+2, is the boundary solid or dashed?
Show a hint
A strict inequality excludes equality.
Reveal answer and explanation
Dashed
Points exactly on y=−x+2 do not satisfy the inequality.
Practice 2
Does (2,6) satisfy the main inequality?
Show a hint
Compare six with twice two plus one.
Reveal answer and explanation
No
6≤5 is false, so the point lies outside the solution region.
Take the idea with you
Interpret a two-variable resource constraint as a region rather than a single line.
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: Recover three quantities from pairwise totals
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