Official Guide Explanation:
Problem Solving #123




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Solution and Metadata

Question: 123
Page: 77
Difficulty: 5 (Moderate)
Category 1: Geometry > Coordinate Geometry > Other
Category 2: Algebra > Inequalities > other
Category 3: Arithmetic > Real Numbers >

Explanation: Before trying sets of coordinates to see if they work in the equation 2x - 3y ≤ -6, think a bit about what values of x and y would result in a value less than -6. For two numbers to add up to a negative, one or both of them has to be negative, so either 2x, -3y, or both must be negative. Put another way, either x must be negative, y must be positive, or both. So, either of the quadrants in which x values are negative (II and III) are possible, as are the quadrants in which y must be positive (I and II). IV, then, is impossible: if x is positive and y is negative, 2x will be positive and -3y will be positive, which cannot sum to a number less than -6.

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