Depth · one concept from first principles · step 2 of 4

Rounding to n means a half-open interval

Let $[a]$ be the integer obtained by rounding $a$ at its tenths digit (to the nearest whole number, with halves rounded up), so $[3.49]=3$ and $[7.5]=8$ . Find the range of values of $x$ that satisfy $2<\left[\frac{x}{4}-1\right]<5$ .

Base-ten numbersExpressions

Your answer

How to solve

Strategy Convert to Algebra — List the possible rounded values, then rewrite "rounds to n" as n-0.5 ≤ t ＜ n+0.5 (picture it on a number line). The rounding disappears and a plain linear inequality is left.

1STEP 1

List the possible rounded values

[x/4-1] is an integer with 2 ＜ [x/4-1] ＜ 5, so it is 3 or 4.

[x/4-1]∈{3, 4}

2STEP 2

Undo the rounding

A number rounds to n exactly when it is at least n-0.5 and less than n+0.5. Join the intervals for 3 and 4 into one.

2.5 ≤ x/4-1 ＜ 4.5

3STEP 3

Solve for x

Add 1 to every part, then multiply every part by 4.

3.5 ≤ x/4 ＜ 5.5 → 14 ≤ x ＜ 22

Answer

14 ≤ x ＜ 22

At x=14: 14/4-1=2.5, which rounds up to 3, allowed. At x=22: 22/4-1=4.5, which rounds to 5, not allowed. At x=13.9: 2.475 rounds to 2, not allowed. So the endpoints are right.

Takeaway

Rounding hides an interval: "rounds to 3" just means "from 2.5 up to, but not including, 3.5".

List the possible rounded values
Undo the rounding
Solve for x

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