Answer:
5 < x < 15
Explanation:
The triangle inequality theorem states that the sum of the measures of any two sides of a triangle must be greater than the measure of the third side
In the given triangle we are provided measures of two of the sides as 10 and 5
Let the measure of the third side be x
So the three sides are 10, 5 and x
Then by the inequality theorem
10 + 5 > x
==> 15 > x or
x < 15 This is an upper bound for x
when we switch sides in an inequality > changes to < and < changes to >
We also have
x + 5 > 10 ==> x > 10 - 5 ==> x > 5
and
x + 10 > 5 ==> x > -5
Since x > 5 is more restrictive than x > -5, we conclude that x > 5 or 5 < x is the lower bound on x
Combining all inequalities we get
5 < x < 15
Note
We could also state the lower and upper bound limits as
difference of two sides < x < sum of two sides
10 - 5 < x < 10 + 5
or
5 < x < 15
While this may seem easier to compute than the explanation given above, the derivation is left out and may confuse some students