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CAT 2021 Slot 1 — QA Question 4

ModulusEasy

The number of integers n that satisfy the inequalities |n - 60| < |n - 100|  < |n - 20| is

Answer & solution

  • A

    18

  • B

    21

  • C

    20

  • 19

Solution

Easy

The critical points 20,60,10020,60,100 split the number line into regions. In each region the absolute values open with definite signs, so the double inequality becomes ordinary linear inequalities — solve region by region and count the integers.

1

Region n≥100n\ge 100. Here ∣n−60∣=n−60|n-60|=n-60 and ∣n−100∣=n−100|n-100|=n-100; compare the first two terms.

\begin{aligned} &|n-60|<|n-100|\\ &\Rightarrow\ n-60
2

Region 60≤n<10060\le n<100. Now ∣n−60∣=n−60|n-60|=n-60, ∣n−100∣=100−n|n-100|=100-n, ∣n−20∣=n−20|n-20|=n-20. Apply both inequalities.

\begin{aligned} &n-60<100-n \Rightarrow 2n<160 \Rightarrow n<80\\ &100-n120 \Rightarrow n>60\\ &\Rightarrow\ 60
3

Region n<60n<60. Here ∣n−100∣=100−n|n-100|=100-n and ∣n−20∣|n-20| — check the second inequality ∣n−100∣<∣n−20∣|n-100|<|n-20| for n<20n<20 first, but the right inequality forces a contradiction with the rest; testing the binding part:

\begin{aligned} &100-n60 \quad\text{(contradicts }n<60\text{)}\\ &\Rightarrow\ \text{no solution in this region} \end{aligned}
4

Count the integers. Only $60 count=79−61+1=19\begin{aligned} &\text{count}=79-61+1=19 \end{aligned}

Number of integers=19(option (d))\text{Number of integers}=19 \quad\text{(option (d))}
CAT 2021 Slot 1 QA Q4: The number of integers n that satisfy the inequalities |n - 60| < |n - 100| < |n - 20| is — Solution | TheCATExam