Depth · one concept from first principles · step 2 of 4
Pin the boundary so exactly three naturals fit
The inequality has exactly solutions that are natural numbers (positive integers). Find the range of values of the constant .
Expressions
Your answer
How to solve
Strategy Convert to Algebra — Turn the inequality into x ≤ (boundary). Then place the boundary on a number line: the naturals below it must be exactly 1, 2, 3.
1STEP 1
Clear the fraction
Multiply both sides by 2 and gather terms.
5x-1-2a ≤ 2x+6 → 3x ≤ 2a+7 → x ≤ (2a+7)/3
Multiplying by a positive number keeps the inequality direction.
7.EE.B.4Convert To Algebra2STEP 2
Place the boundary on a number line
For the natural solutions to be exactly 1, 2, 3, the boundary must reach 3 (3 included) but stay below 4 (4 excluded).
3 ≤ (2a+7)/3 ＜ 4
The boundary may equal 3 because the sign is ≤ ; it may not equal 4, or 4 would also be a solution.
6.EE.B.8Draw A DiagramThe inequality has exactly three natural-number solutions precisely when .
Why?
If a whole number is at or below the boundary , every smaller whole number is below it too, so the natural solutions always form a run with no gaps.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
Why?
A run starting at matches up with exactly three counters when is in it and is not, so must be at most the boundary and must be beyond it.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Solve for a
Multiply every part by 3, subtract 7, divide by 2.
9 ≤ 2a+7 ＜ 12 → 2 ≤ 2a ＜ 5 → 1 ≤ a ＜ 5/2
Applying the same operation to all three parts keeps the chain true.
7.EE.B.4Convert To AlgebraAnswer
1 ≤ a ＜ 5/2
At a=1 the solution set is x ≤ 3: naturals 1, 2, 3, exactly three. At a=5/2 it is x ≤ 4, giving four naturals, so 5/2 must be excluded. At a=0.9 it is x ≤ 8.8/3, about 2.93, only two naturals.
Takeaway
Counting solutions is really about where the boundary sits between two whole numbers.
- Clear the fraction
- Place the boundary on a number line
- Solve for a
Where next?
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