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



Answer :

GinnyAnswer
To determine the range of possible values for [tex]\( x \)[/tex], the length of the third side of the triangle, we 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 third side.

Given sides are 2 inches and 7 inches. Let [tex]\( x \)[/tex] be the third side. We need to satisfy the following three conditions based on the triangle inequality theorem:

1. [tex]\( x + 2 > 7 \)[/tex] (sum of [tex]\( x \)[/tex] and 2 greater than 7)
2. [tex]\( x + 7 > 2 \)[/tex] (sum of [tex]\( x \)[/tex] and 7 greater than 2)
3. [tex]\( 2 + 7 > x \)[/tex] (sum of 2 and 7 greater than [tex]\( x \)[/tex])

Let's solve each inequality:

1. [tex]\( x + 2 > 7 \)[/tex]
[tex]\[ x > 5 \][/tex]

2. [tex]\( x + 7 > 2 \)[/tex]
[tex]\[ x > -5 \][/tex]

Since [tex]\( x \)[/tex] is a length, it must be positive, and [tex]\( x > -5 \)[/tex] is always true for any positive value of [tex]\( x \)[/tex].

3. [tex]\( 2 + 7 > x \)[/tex]
[tex]\[ 9 > x \][/tex]
[tex]\[ x < 9 \][/tex]

Combining these inequalities, we get:
[tex]\[ 5 < x < 9 \][/tex]

Therefore, the range of possible values for [tex]\( x \)[/tex] is given by the inequality [tex]\( 5 < x < 9 \)[/tex].

The correct answer is [tex]\( \boxed{A} \)[/tex].

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