Find the range of possible measures if each set of expressions represents measures of the sides of a triangle:x 2x+10, x+8
To find the range of possible measures for the sides of a triangle using the given expressions, we need to consider the triangle inequality theorem. According to this theorem, in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
In this case, we have three expressions representing the measures of the sides: x, 2x+10, and x+8. Let's consider different scenarios for the values of x:
1. When x is the smallest side:
- The sum of the other two sides should be greater than the smallest side.
- (2x+10) + (x+8) > x
- 3x + 18 > x
- 2x > -18
- x > -9
Therefore, there is no lower bound for the value of x.
2. When x+8 is the smallest side:
- The sum of the other two sides should be greater than the smallest side.
- x + (2x+10) > x+8
- 3x + 10 > x + 8
- 2x > -2
- x > -1
Therefore, the lower bound for the value of x is -1.
3. When 2x+10 is the smallest side:
- The sum of the other two sides should be greater than the smallest side.
- x + (x+8) > 2x + 10
- 2x + 8 > 2x + 10
- 8 > 10
This is not a valid scenario since 8 is not greater than 10.
In summary, the range of possible measures for x is x > -1.