The output is twice the input. Answer in this form y=mx+b

  b = 2, GH = 9

Step-by-step explanation:

The last two equations tell you ...

  HI = 3 = 4b -5

  8 = 4b . . . . . . . . . add 5

  2 = b . . . . . . . . . . divide by 4

Then ...

  GI = 5·2 +2 = 12

So ...

  GI/HI = 12/3 = 4 = G/H . . . . use the found value for GI and the given HI

  G = 4H . . . . . . . multiply by H

  GH = 4H² . . . . . multiply by H again

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At this point, it can be useful to try the offered solutions to see what works.

For GH = 9

  GH = 4H² = 9

  H = ±√(9/4) = ±3/2

  G = 4H = ±6

Then ...

  I = 3/H = 3/(±3/2) = ±2 . . . . matching the sign of H

We want values of G and H such that H is between G and I.

  positive H: (G, H, I) = (6, 3/2, 2/3) . . . . yes, H is between G and I

  negative H: (G, H, I) = (-6, -3/2, -2/3) . . yes, H is between G and I

The solution b=2, GH=9 satisfies problem requirements.

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For GH = 12

  GH = 4H² = 12

  H = ±√(12/4) = ±√3

  G = 4H = ±4√3

  I = 3/H = 3/±√3 = ±√3 . . . . sign matches H

Then (G, H, I) = (4√3, √3, √3) . . . . . H = I; is not "between" G and I

or (G, H, I) = (-4√3, -√3, -√3) . . . . . . H = I; is not "between" G and I

The solution b=2, GH=12 will only satisfy the problem requirements if you interpret "between G and I" to allow that H = I.