CAT 2021 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like **Speed-time-distance**, **Mixture and Alligations**. This year, there was a surprise. The questions from **Geometry** were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 16 : For all real numbers x the condition |3x - 20| + |3x - 40| = 20 necessarily holds if

- 6 < x < 11
- 7 < x < 12
- 10 < x < 15
- 9 < x < 14

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Let’s assume y to be 3x,

Now, the equation becomes | y - 20 | + | y - 40 | = 20

Let’s comprehend the meaning of | y - 20 |,

| y - 20 | simply means, the distance between y and 20

Let’s jot the three terms, y, 20 and 40 on a number line.

Now, y could be anywhere on the number line.

Observe that if y is on the left of 20, the distance between y and ‘40’ will be greater than 20 and the condition | y - 20 | + | y - 40 | = 20 fails.

If y is on the right of 40, the distance between y and ‘20’ will be greater than 20 and the condition | y - 20 | + | y - 40 | = 20 fails.

Hence the point y needs to be between 20 and 40.

Will any point between 20 and 40 qualify to be ‘y’ ?

Yes, Since the distance between 20 and 40 is exactly 20, for any ‘y’ between 20 and 40, the distance between 20 and ‘y’ plus the distance between ‘y’ and 40 is exactly 20.

So, 20 < y < 40

20 < 3x < 40

20/3 < x < 40/3

Option B alone holds good.

__Alternate Solution:__

We know that the minimum value of the modulus function can be zero.

Thus value occurs in the two modulus functions at 20/3 and 40/3 respectively.

So, for numbers less than 20/3 the value of both the functions increases and similarly, the value of the sum of the functions increases when x is greater than 40/3.

Between 20/3 and 40/3, the value of the functions remains the same as one of the inequality increases by the same amount as the decrease in the other.

So, the value of the sum of the modulus remains at 20 for the range of values [20/3 , 40/3].

Which can be rewritten as [6.66,13.33]

The option which satisfies this condition is 7 < x < 12.

The question is **" For all real numbers x the condition |3x - 20| + |3x - 40| = 20 necessarily holds if " **

Choice B is the correct answer.

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