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In triangle FAB , FA = 2 x + 7, AB = x + 12, and FB = 4 x .

What must be true about the value of x ?

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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.

User Matt Austin
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