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]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 the third side [tex]\( x \)[/tex] in a triangle with sides measuring 2 inches and 7 inches, we need to use the triangle inequality theorem. The triangle inequality 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:
- Side1 = 2 inches
- Side2 = 7 inches
- Side3 = [tex]\( x \)[/tex] inches

We need to ensure the following inequalities hold true:

1. [tex]\( \text{Side1} + \text{Side2} > \text{Side3} \)[/tex]
[tex]\[ 2 + 7 > x \][/tex]
[tex]\[ 9 > x \][/tex]
[tex]\[ x < 9 \][/tex]

2. [tex]\( \text{Side1} + \text{Side3} > \text{Side2} \)[/tex]
[tex]\[ 2 + x > 7 \][/tex]
[tex]\[ x > 5 \][/tex]

3. [tex]\( \text{Side2} + \text{Side3} > \text{Side1} \)[/tex]
[tex]\[ 7 + x > 2 \][/tex]
[tex]\[ x > -5 \][/tex]
Since [tex]\( x \)[/tex] represents a length, this condition [tex]\( x > -5 \)[/tex] is always true and does not provide new information for positive values of [tex]\( x \)[/tex].

Combining the two meaningful 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]

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

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