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
- Apply the same operation to every part of a compound inequality.
- 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
Solve 5 ≤ 2x + 1 < 13 over the real numbers.
Why this math matters
Represent measurements that must remain within lower and upper tolerances. 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.
Solve a two-sided bound
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve 5 ≤ 2x + 1 < 13 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
4 ≤ 2x < 12
Subtract one from all three parts.
Develop the calculation
2 ≤ x < 6
Divide every part by the positive factor two.
Check and interpret
Solution interval: [2,6)
The lower endpoint is included and the upper endpoint is excluded.
The result
Solution interval: [2,6)
The lower endpoint is included and the upper endpoint is excluded.
Common mistakes to catch
- Operate on both bounds as well as the middle expression.
- An 'and' compound bound requires both conditions simultaneously.
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
Which integers satisfy the main bound?
Show a hint
List whole integers in the interval.
Reveal answer and explanation
2, 3, 4, 5
Six is excluded by the strict upper inequality.
Practice 2
Solve −1 < y−3 ≤ 4.
Show a hint
Add three to all parts.
Reveal answer and explanation
2 < y ≤ 7
The endpoint inclusion types stay unchanged under translation.
Take the idea with you
Represent measurements that must remain within lower and upper tolerances.
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 a distance equation with two directions
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