M26-12

Official Solution:

If \(-3 < x < 5\) and \(-7 < y < 9\), which of the following represents the range of all possible values of \(y-x\)?

A. \(-4 \lt y-x \lt 4\)
B. \(-2 \lt y-x \lt 4\)
C. \(-12 \lt y-x \lt 4\)
D. \(-12 \lt y-x \lt 12\)
E. \(4 \lt y-x \lt 12\)


To obtain the maximum value of \(y-x\), consider the maximum value of \(y\) and the minimum value of \(x\): \(9-(-3)=12\);

To obtain the minimum value of \(y-x\), consider the minimum value of \(y\) and the maximum value of \(x\): \(-7-(5)=-12\);

Therefore, the range of all possible values of \(y-x\) lies between \(-12\) and \(12\): \(-12 < y-x < 12\).

Alternatively, we can approach the problem in the following way. We know that \(-3 < x < 5\) and \(-7 < y < 9\).

Add \(y < 9\) to \(-3 < x\) to get \(y - 3 < 9 + x\), which can be rearranged as \(y - x < 12\). This gives the upper bound for the value of \(y-x\).

Add \(-7 < y\) to \(x < 5\) to get \(-7 + x < y + 5\), which can be rearranged as \(-12 < y - x\). This gives the lower bound for the value of \(y-x\).

Thus, we establish that \(-12 < y - x < 12\).


Answer: D