To determine the possible range of values for the remaining side [tex]\( x \)[/tex] in a triangle where two sides measure 15 inches and 7 inches, we can 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. Let's use this theorem to find the constraints on [tex]\( x \)[/tex].
1. According to the triangle inequality theorem, the following conditions must be satisfied:
[tex]\[
x + 7 > 15
\][/tex]
[tex]\[
x + 15 > 7
\][/tex]
[tex]\[
15 + 7 > x
\][/tex]
2. Let's solve each inequality for [tex]\( x \)[/tex]:
- From [tex]\( x + 7 > 15 \)[/tex]:
[tex]\[
x > 15 - 7
\][/tex]
[tex]\[
x > 8
\][/tex]
- From [tex]\( x + 15 > 7 \)[/tex]:
[tex]\[
x > 7 - 15
\][/tex]
Since [tex]\( 7 - 15 \)[/tex] is [tex]\(-8\)[/tex], this condition, [tex]\( x > -8 \)[/tex], is automatically satisfied because all sides of a triangle must be positive, i.e., [tex]\( x \)[/tex] must be greater than zero, which is always true for [tex]\( x > 8 \)[/tex].
- From [tex]\( 15 + 7 > x \)[/tex]:
[tex]\[
22 > x
\][/tex]
[tex]\[
x < 22
\][/tex]
3. Combining the inequalities, we get the range for [tex]\( x \)[/tex]:
[tex]\[
8 < x < 22
\][/tex]
Therefore, the possible range of values for the remaining side [tex]\( x \)[/tex] in the triangle is:
[tex]\[
8 < x < 22
\][/tex]
Given these constraints, the correct answers are:
- [tex]\( x < 22 \)[/tex]
- [tex]\( x > 8 \)[/tex]
Hence, the correct options are [tex]\( x < 22 \)[/tex] and [tex]\( x > 8 \)[/tex].
