TheCATExam
Sign in

CAT 2024 Slot 2 — QA Question 22

ModulusEasy

If x and y satisfy the equations |x| + x + y = 15 and x + |y| - y = 20, then (x - y) equals

Answer & solution

Correct answer: 15

  • 15

  • B

    10

  • C

    20

  • D

    5

Solution

Medium

Each ∣⋅∣|\cdot| collapses depending on the sign of xx and yy. The expressions ∣x∣+x|x|+x and ∣y∣−y|y|-y are zero or doubled, so test the sign case that keeps both equations consistent.

1

Try x≥0x\ge0 and y<0y<0. Then ∣x∣+x=2x|x|+x=2x and ∣y∣−y=−2y|y|-y=-2y.

2x+y=15(from ∣x∣+x+y=15)x−2y=20(from x+∣y∣−y=20)\begin{aligned} &2x+y=15 \quad\text{(from }|x|+x+y=15\text{)}\\ &x-2y=20 \quad\text{(from }x+|y|-y=20\text{)} \end{aligned}
2

Solve the linear system. From the first, y=15−2xy=15-2x; substitute into the second.

x−2(15−2x)=20⇒ x−30+4x=20(expand)⇒ 5x=50 ⇒ x=10,  y=15−20=−5\begin{aligned} &x-2(15-2x)=20\\ &\Rightarrow\ x-30+4x=20 \quad\text{(expand)}\\ &\Rightarrow\ 5x=50\ \Rightarrow\ x=10,\ \ y=15-20=-5 \end{aligned}
3

Check signs and compute. x=10≥0x=10\ge0 and y=−5<0y=-5<0 match the assumed case, so it is valid.

x−y=10−(−5)=15\begin{aligned} &x-y=10-(-5)=15 \end{aligned}
x−y=15x-y=15

Related Modulus questions

See all Inequalities & Modulus questions →
CAT 2024 Slot 2 QA Q22 — Modulus solution | TheCATExam