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If I x + 1 I ≤ 9 and I y - 1 I ≤ 7 then a ≤ 3y - 2x ≤ b . What is the value of b - a ?

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by Souvik118

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by Souvik118

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| X + 1 | <= 9

X + 1 <= 9 => X <= 8

- (X + 1) <= 9 => X >= -10

| Y - 1 | <= 7

Y+1 <= 7 and -Y+1 <= 7 => - 6 <= Y <= 8

So draw coordinate axes and draw two lines at X = -10, and 8.

Then draw two lines at Y = -6 and 8.

You get a rectangle, with corners A(-10, 8), B(8, 8), C(8, -6) and D(-10, -6).

Expression : let 3 Y - 2 X = K then Y = 2X/3 + K/3

This is a line with positive slope. By varying K, we can place this line. It will enter the rectangular region ABCD at A(-10,8) and as you move the straight line to the right, it will leave the ABCD at C (8, -6).

Hence at A, K = 3 * 8 - 2 *(-10) = 44

at C, K = 3 *(-6) - 2 * 8 = - 34

So, -34 <= 3Y - 2X <= 44

a = -34 and b = 44

Hence, b - a = 78

You can draw a diagram with coordinate axes and see. Then it will be easier to understand.

X + 1 <= 9 => X <= 8

- (X + 1) <= 9 => X >= -10

| Y - 1 | <= 7

Y+1 <= 7 and -Y+1 <= 7 => - 6 <= Y <= 8

So draw coordinate axes and draw two lines at X = -10, and 8.

Then draw two lines at Y = -6 and 8.

You get a rectangle, with corners A(-10, 8), B(8, 8), C(8, -6) and D(-10, -6).

Expression : let 3 Y - 2 X = K then Y = 2X/3 + K/3

This is a line with positive slope. By varying K, we can place this line. It will enter the rectangular region ABCD at A(-10,8) and as you move the straight line to the right, it will leave the ABCD at C (8, -6).

Hence at A, K = 3 * 8 - 2 *(-10) = 44

at C, K = 3 *(-6) - 2 * 8 = - 34

So, -34 <= 3Y - 2X <= 44

a = -34 and b = 44

Hence, b - a = 78

You can draw a diagram with coordinate axes and see. Then it will be easier to understand.