We are given three expressions for measures of the sides of a triangle.
x, 2x+1, x+4
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. Using this theorem, we can write three true inequalities for this triangle.
(I)& x+ 2x+1> x+4
(II)& x+ x+4> 2x+1
(III)& 2x+1+ x+4> xLet's solve each inequality for x and present it using the following table.
Inequality
Simplified
Solution Set
x+2x+1>x+4
x+x+4>2x+1
2x+1+x+4>x
Let's simplify each inequality.
Inequality
Simplified
Solution Set
x+2x+1>x+4
2x>3
x+x+4>2x+1
4>1
2x+1+x+4>x
2x>- 5
Next, write the solution set into the table.
Inequality
Simplified
Solution Set
x+2x+1>x+4
2x>3
x>1.5
x+x+4>2x+1
4>1
All solutions
2x+1+x+4>x
2x>- 5
x>- 2.5
Notice that in the second inequality, we found that 4 is always greater than 1. Therefore, all solutions satisfy this inequality.
Let's graph the other two inequalities and find the set of common solutions.
As we can see, the graphs of each solut set overlap after 1.5. Therefore, the possible values of x are the following.
x>1.5
