GMAT Number Properties Explained — 2 x+y=12 |y| \leq 12 For how many ordered pairs (x, y)...

2 x + y = 12 ∣ y ∣ \leq 12 For how many ordered pairs (x, y) that are solutions of the system above are x and y both integers?

inequalitiessimultaneous equationsnumber theory

Explanation	Calculations	Help
$We first isolate y in the first equation, in order to use substitution into the absolute value inequality.$	$y = 12 - 2x$
$We apply the constraint on y by substituting its expression into the absolute value inequality.$	$\|12 - 2x\| \leq 12$	Theory & Tactics Method Card EQUA3-A Click to view full details
$We convert the absolute value inequality into a compound inequality.$	$-12 \leq 12 - 2x \leq 12$	Theory & Tactics Method Card INEQ3 Click to view full details
$Statement: Solve the left part of the compound inequality for x by isolating x .$	$-12 \leq 12 - 2x -24 \leq -2x 12 \geq x$	Theory & Tactics Method Card INEQ1-B Click to view full details
$Statement: Solve the right part of the compound inequality for x by isolating x .$	$12 - 2x \leq 12 -2x \leq 0 x \geq 0$	Theory & Tactics Method Card INEQ1-B Click to view full details
$We combine the two results to find the range for x .$	$0 \leq x \leq 12$
$We count the integer values from 0 to 12 inclusive to find the number of possible x values.$	$12 - 0 + 1 = 13$	Theory & Tactics Method Card EVEN1-A Click to view full details

Scroll horizontally to view all columns

13