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Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Translate inclusive boundaries into symbols.
- Apply operations to all parts of an inequality.
- List the feasible integer counts.
Before you start
Solve linear inequalities with positive coefficients and interpret ≤.
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
An empty classroom package has mass 50 g. Each identical booklet adds 25 g. A fictional sorting rule accepts packages from 250 g through 350 g, inclusive. How many booklets are allowed?
Why this math matters
A permitted range has two boundaries, and a valid answer must respect both. A compound inequality keeps them together. The algebra describes an interval, while the physical objects limit the possible inputs to whole counts.
Set up the model
A useful answer starts with clear assumptions:
- Every booklet has exactly the stated mass in the model.
- Only complete booklets are packed, and the fictional boundaries include both endpoints.
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.
Find every count that fits a package-mass range
PausedQuestion: Start with the question. Paused.
Question
Start with the question
An empty classroom package has mass 50 g. Each identical booklet adds 25 g. A fictional sorting rule accepts packages from 250 g through 350 g, inclusive. How many booklets are allowed?
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.
Model the allowed total
250 ≤ 50 + 25n ≤ 350
The package mass must be at least 250 g and at most 350 g. Both conditions apply to the same count n.
Remove the empty package
200 ≤ 25n ≤ 300
Subtract fifty from all three parts. This expresses the allowed mass contributed by booklets alone.
Convert mass to counts
8 ≤ n ≤ 12; n ∈ {8, 9, 10, 11, 12}
Divide all parts by positive twenty-five, so the inequality directions stay unchanged. Keep only whole counts in the resulting interval.
The result
The package may contain 8, 9, 10, 11, or 12 booklets.
The algebraic real-number interval is [8, 12], but 8.5 booklets is excluded by the context. Checking the boundary counts gives total masses of 250 g and 350 g exactly.
Common mistakes to catch
- Forgetting the empty package allows too many booklets.
- Treating 'through 350 inclusive' as 'less than 350' wrongly removes the upper endpoint.
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
An empty box is 100 g and each item is 40 g. Which whole counts give totals from 260 g through 340 g?
Show a hint
Subtract one hundred, then divide by forty.
Reveal answer and explanation
4, 5, or 6 items
260 ≤ 100 + 40n ≤ 340 becomes 160 ≤ 40n ≤ 240, so 4 ≤ n ≤ 6.
Practice 2
In the original booklet problem, change the upper rule to strictly less than 350 g. Which counts remain?
Show a hint
Only the upper boundary's inclusion changes.
Reveal answer and explanation
8, 9, 10, or 11 booklets
The new condition is 8 ≤ n < 12. Twelve gives exactly 350 g, which is now excluded.
Take the idea with you
Underline words such as 'at least', 'less than', and 'inclusive' before choosing inequality signs.
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.
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