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

This gives us three inequalities to consider:
1. [tex]\(200 + 300 > x\)[/tex]
2. [tex]\(200 + x > 300\)[/tex]
3. [tex]\(300 + x > 200\)[/tex]

Let’s explore each inequality individually:

1. From the first inequality:
[tex]\[ 200 + 300 > x \implies 500 > x \implies x < 500 \][/tex]

2. From the second inequality:
[tex]\[ 200 + x > 300 \implies x > 300 - 200 \implies x > 100 \][/tex]

3. From the third inequality:
[tex]\[ 300 + x > 200 \implies x > 200 - 300 \implies x > -100 \][/tex]

However, since side lengths of a triangle cannot be negative, we ignore [tex]\( x > -100 \)[/tex] as it’s always satisfied if [tex]\( x > 100 \)[/tex].

Combining the viable conditions from the inequalities, we get:
[tex]\[ 100 < x < 500 \][/tex]

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