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\(y < x - a\) \(y > x + b\)In the xy-plane, if (0, 0) is a solution to t...


Question

\(y < x - a\)
\(y > x + b\)

In the xy-plane, if (0, 0) is a solution to the system of inequalities above, which of the following relationships between a and b must be true?

 

Options

A)
a > b
B)
b > a
C)
|a| > |b|
D)
a = -b

The correct answer is A.

Explanation:

Choice A is correct. Since (0,0) is a solution to the system of inequalities, substituting 0 for x and 0 for y in the given system must result in two true inequalities. After this substitution, \(y < -x + a\) becomes \(0 < a\), and \(y > x + b\) becomes \(0 > b\). Hence, \(a\) is positive and \(b\) is negative. Therefore, \(a > b\).
Choice B is incorrect because \(b > a\) cannot be true if \(b\) is negative and \(a\) is positive. Choice C is incorrect because it is possible to find an example where (0,0) is a solution to the system, but \(|a| < |b|\);  for example, if \(a = 6\) and \(b = -7\). Choice D is incorrect because the equation \(a = -b\) doesn't have to be true;  for example, (0,0) is a solution to the system of inequalities if \(a = 1\) and \(b = -2\).


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