How do i write and solve for this inequality

9514 1404 393

  -7/4 < x < 8

Step-by-step explanation:

The value of y can be determined from the sum of the angles, so you know each of the angles exactly. That means you know the ratio of side lengths exactly, which lets you solve for x exactly.

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Setting that aside, we observe that angle C is greater than angle A, so side AB will be longer than side BC.

  3x +15 > 4x +7

  8 > x . . . . . . . . . . subtract 3x+7

We also know that the lengths of these sides must be positive. Since BC is the shorter side, we require ...

  4x +7 > 0

  4x > -7

  x > -7/4

So, the allowable values of x are ...

  -7/4 < x < 8

More complete solution

If we read the figure correctly, the sum of angles is ...

  (2y +12) +(4y +12) +(y -18) = 180

  7y +6 = 180

  y = (180 -6)/7 = 24 6/7°

Then (in degrees) ...

  ∠A = 2(24 6/7) +12 = 61 5/7, and ∠C = 4(24 6/7) +12 = 111 3/7

The Law of Sines tells us ...

  AB/sin(C) = BC/sin(A)

  sin(A)(3x +15) = sin(C)(4x+7)

  x(4sin(C) -3sin(A)) = 15sin(A) -7sin(C)

  x = (15sin(A) -7sin(C))/(4sin(C) -3sin(A))

  x ≈ 6.1872652

where are the options?

step-by-step explanation: