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Exercise 2.10 · Q1

Q.x≤3y, x≥yx\le3y,\ x\ge y.

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✓ Free question

Step 1. Both boundaries, x=3yx=3y (i.e. y=x3y=\frac x3) and x=yx=y, pass through the origin, so test another point instead, e.g. (2,1)(2,1): x≤3y⇒2≤3x\le3y\Rightarrow2\le3 ✓; x≥y⇒2≥1x\ge y\Rightarrow2\ge1 ✓ -- so (2,1)(2,1) lies in the required region.

Step 2. For x<0x<0 a point such as (−2,−1)(-2,-1) gives x≤3y⇒−2≤−3x\le3y\Rightarrow-2\le-3, which is FALSE, so the negative side is excluded.

Step 3. The feasible region is the open wedge with vertex at the origin, lying between the ray y=xy=x (upper boundary) and the ray y=x3y=\dfrac x3 (lower boundary), for x≥0x\ge0: x3≤y≤x\dfrac x3\le y\le x.

✓Final answer

The wedge between y=x3y=\dfrac x3 and y=xy=x for x≥0x\ge0 (both boundary lines included).

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