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A triangle has side lengths of 200 units and 300 units. Write a compound inequality for the range of the possible lengths for the third side, [tex]\( x \)[/tex].

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To determine the possible lengths for the third side [tex]\( x \)[/tex] of a triangle with given sides 200 units and 300 units, 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 remaining side.

Given the sides [tex]\( a = 200 \)[/tex] units and [tex]\( b = 300 \)[/tex] units, we consider the following inequalities:

1. [tex]\( a + b > x \)[/tex]
2. [tex]\( a + x > b \)[/tex]
3. [tex]\( b + x > a \)[/tex]

Now, we can simplify these inequalities:

1. Since [tex]\( a = 200 \)[/tex] and [tex]\( b = 300 \)[/tex]:
[tex]\[ 200 + 300 > x \implies 500 > x \][/tex]

2. Since [tex]\( a = 200 \)[/tex] and [tex]\( b = 300 \)[/tex]:
[tex]\[ 200 + x > 300 \implies x > 300 - 200 \implies x > 100 \][/tex]

3. Since [tex]\( a = 200 \)[/tex] and [tex]\( b = 300 \)[/tex]:
[tex]\[ 300 + x > 200 \][/tex]

The third inequality [tex]\( 300 + x > 200 \)[/tex] is always true for any positive [tex]\( x \)[/tex] as long as [tex]\( x \)[/tex] satisfies the first two inequalities. So, we can put that aside.

Combining the simplified inequalities from steps 1 and 2 gives us the compound inequality:
[tex]\[ 100 < x < 500 \][/tex]

Therefore, the range of possible lengths for the third side, [tex]\( x \)[/tex], is:
[tex]\[ 100 < x < 500 \][/tex]
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