Answered

A triangle has sides measuring 2 inches and 7 inches. If [tex]\( x \)[/tex] represents the length in inches of the third side, which inequality gives the range of possible values for [tex]\( x \)[/tex]?

A. [tex]\( 2 \ \textless \ x \ \textless \ 7 \)[/tex]
B. [tex]\( 5 \leq x \leq 9 \)[/tex]
C. [tex]\( 5 \ \textless \ x \ \textless \ 9 \)[/tex]
D. [tex]\( 2 \leq x \leq 7 \)[/tex]



Answer :

GinnyAnswer
To determine the possible values for the third side [tex]\( x \)[/tex] of a triangle with given sides measuring 2 inches and 7 inches, we will use the triangle inequality theorem. This 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 sides:
1. [tex]\( \text{side}_1 = 2 \)[/tex] inches
2. [tex]\( \text{side}_2 = 7 \)[/tex] inches

We'll let [tex]\( x \)[/tex] be the length of the third side. The triangle inequality theorem gives us three conditions:

1. [tex]\( \text{side}_1 + \text{side}_2 > x \)[/tex]:
[tex]\[ 2 + 7 > x \][/tex]
[tex]\[ 9 > x \quad \text{(or equivalently, } x < 9) \][/tex]

2. [tex]\( \text{side}_1 + x > \text{side}_2 \)[/tex]:
[tex]\[ 2 + x > 7 \][/tex]
[tex]\[ x > 5 \][/tex]

3. [tex]\( \text{side}_2 + x > \text{side}_1 \)[/tex]:
[tex]\[ 7 + x > 2 \][/tex]
[tex]\[ x > -5 \quad \text{(This is always true since \( x \) must be positive)} \][/tex]

The more stringent conditions derived from the triangle inequality theorem are:
[tex]\[ 5 < x < 9 \][/tex]

Therefore, the correct inequality that describes the range of possible values for [tex]\( x \)[/tex] is:

[tex]\[ 5 < x < 9 \][/tex]

So, the correct answer is:
C. [tex]\( 5 < x < 9 \)[/tex]

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