Depth · one concept from first principles · step 2 of 4
Solve each line with the curve in the first quadrant.
So the first-quadrant region runs from x = 0 to x = 8.
7.RP.A.2Identify SubproblemsFor an integer x from 1 to 8, y must satisfy x/4 ≤ y ≤ 4x and y ≤ 16/x.
The lower edge is the line y=(1/4)x; the upper edge is y=4x until A, then the curve.
7.RP.A.2Make A Systematic ListCount y in each column: x=1 gives 4, x=2 gives 8, x=3 gives 5, x=4 gives 4, x=5 gives 2, and x=6, 7, 8 give 1 each.
For example at x=5: y ≥ 1.25 and y ≤ 3.2, so y = 2, 3.
6.NS.C.8Make A Systematic ListThe graphs are symmetric about the origin, so the third quadrant mirrors these 26 points; the shared origin counts once.
(x, y) is in one region exactly when (-x, -y) is in the other.
6.NS.C.8Visualize Spatial RelationshipsThe third-quadrant region holds exactly as many lattice points as the first, so with the origin the total is .
A half turn about O, sending to , carries the first region exactly onto the second.
Multiplying both sides of or by keeps them equal, so each line is unchanged by the turn.
is a product of like signs, equal to , so stays true and the curve is unchanged too.
The turn pairs each lattice point of the first region with exactly one of the second, so the second region also holds .
Points with , points with , and the single point O on never overlap and cover everything, so their counts add: .
Symmetry halves the counting; a column-by-column list keeps it honest.
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