Answer:
Explanation:
To determine what must be true about the value of x in triangle FAB, we can use the fact that in a triangle, the sum of the lengths of any two sides is always greater than the length of the third side.
In this case, we have FA = 2x + 7, AB = x + 12, and FB = 4x. According to the triangle inequality theorem, we can write the following inequalities:
FA + AB > FB,
FA + FB > AB,
AB + FB > FA.
Substituting the given values, we get:
(2x + 7) + (x + 12) > 4x,
(2x + 7) + 4x > (x + 12),
(x + 12) + 4x > (2x + 7).
Simplifying these inequalities, we have:
3x + 19 > 4x,
6x + 7 > x + 12,
5x + 12 > 2x + 7.
Now, let's solve each inequality to find the range of possible values for x:
3x + 19 > 4x:
Subtracting 3x from both sides: 19 > x.
6x + 7 > x + 12:
Subtracting x from both sides: 5x + 7 > 12.
Subtracting 7 from both sides: 5x > 5.
Dividing both sides by 5: x > 1.
5x + 12 > 2x + 7:
Subtracting 2x from both sides: 3x + 12 > 7.
Subtracting 12 from both sides: 3x > -5.
Dividing both sides by 3: x > -5/3.
Combining the results, we find that x must satisfy the following conditions:
x > 1 (from 6x + 7 > x + 12),
x > -5/3 (from 5x + 12 > 2x + 7),
x > 19 (from 3x + 19 > 4x).
Therefore, the value of x must be greater than 19 in order for the given triangle FAB to exist.