To determine the possible values of [tex]\(x\)[/tex] that satisfy the triangle inequalities for the side lengths represented by the expressions [tex]\(3x + 1\)[/tex], [tex]\(-x - 4\)[/tex], and [tex]\(3x - 6\)[/tex], we will use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Given this, we need to check the following inequalities:
1. [tex]\( (3x + 1) + (-x - 4) > 3x - 6 \)[/tex]
2. [tex]\( (3x + 1) + (3x - 6) > -x - 4 \)[/tex]
3. [tex]\( (-x - 4) + (3x - 6) > 3x + 1 \)[/tex]
We'll analyze each inequality one by one:
### Inequality 1:
[tex]\[ (3x + 1) + (-x - 4) > 3x - 6 \][/tex]
Simplify the left side:
[tex]\[ 3x - x + 1 - 4 > 3x - 6 \][/tex]
[tex]\[ 2x - 3 > 3x - 6 \][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[ -3 > x - 6 \][/tex]
Add 6 to both sides:
[tex]\[ 3 > x \][/tex]
This can be written as:
[tex]\[ x < 3 \][/tex]
### Inequality 2:
[tex]\[ (3x + 1) + (3x - 6) > -x - 4 \][/tex]
Simplify the left side:
[tex]\[ 3x + 3x + 1 - 6 > -x - 4 \][/tex]
[tex]\[ 6x - 5 > -x - 4 \][/tex]
Add [tex]\(x\)[/tex] to both sides:
[tex]\[ 7x - 5 > -4 \][/tex]
Add 5 to both sides:
[tex]\[ 7x > 1 \][/tex]
Divide by 7:
[tex]\[ x > \frac{1}{7} \][/tex]
### Inequality 3:
[tex]\[ (-x - 4) + (3x - 6) > 3x + 1 \][/tex]
Simplify the left side:
[tex]\[ -x + 3x - 4 - 6 > 3x + 1 \][/tex]
[tex]\[ 2x - 10 > 3x + 1 \][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[ -10 > x + 1 \][/tex]
Subtract 1 from both sides:
[tex]\[ -11 > x \][/tex]
This can be written as:
[tex]\[ x < -11 \][/tex]
We combine the valid ranges from the first and second inequalities:
1. [tex]\(x < 3\)[/tex]
2. [tex]\(x > \frac{1}{7}\)[/tex]
The third inequality [tex]\(x < -11\)[/tex] is not consistent with the other inequalities, so it cannot be satisfied unless there is no mutual solution.
By analyzing and combining the constraints [tex]\(x < 3\)[/tex] and [tex]\(x > \frac{1}{7}\)[/tex], we typically find:
[tex]\[ \frac{1}{7} < x < 3 \][/tex]
Given the result:
The correct answer is:
```
False
```
The solution reveals that no real number can satisfy all constraints simultaneously. Hence, the side lengths given do not form a valid triangle for any [tex]\(x\)[/tex]. Hence, the provided options would not represent valid ranges for [tex]\(x\)[/tex].
