To begin writing the inequalities, we can choose any two sides of the triangle and add them together. We will arbitrarily choose to add 7 and x for the first inequality. The sum of these should be larger than the third side.
7+x>5
The other two inequalities can be written in a similar way. The table below shows all three inequalities.
#
Inequality
1
7+x>5
2
x+5>7
3
5+7>x
To determine if it is possible for x to equal 1 we will solve all three inequalities. Let's start with the first one.
We will solve the other two inequalities in a similar way.
#
Inequality
Solution
1
7+x>5
x>- 2
2
x+5>7
x>2
3
5+7>x
x<12
The table above shows that x must be greater than - 2, x must be greater than 2, and x must be less than 12. All of these apply at the same time, which means that the solution from Inequality 2 overrides the solution to Inequality 1. This means that the possible range of values for x is between 2 and 12, exclusive.
2
