### Instruction: A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order.### Response: I know that a triangle can only exist if the sum of any two sides is greater than the third side.
So I can use this condition to find some inequalities that $x^2$ must satisfy.
First, I consider the pair of sides $7$ and $10$.
The sum of these sides must be greater than $x^2$, so I have the inequality $7 + 10 > x^2$, or $17 > x^2$.
Next, I consider the pair of sides $7$ and $x^2$.
The sum of these sides must be greater than $10$, so I have the inequality $7 + x^2 > 10$, or $x^2 > 3$.
Finally, I consider the pair of sides $10$ and $x^2$.
The sum of these sides must be greater than $7$, so I have the inequality $10 + x^2 > 7$, or $x^2 > -3$.
Since $x$ is a positive integer, I can ignore the last inequality, as it is always true.
So I only need to focus on the first two inequalities, $x^2 > 3$ and $17 > x^2$.
To find the positive integer values of $x$ that satisfy both inequalities, I can use a number line to visualize the possible values of $x^2$.
I mark the points $3$ and $17$ on the number line, and I shade the region between them, as that is where $x^2$ must lie.
Then I look for the perfect squares that fall within this region.
I see that $4$, $9$, and $16$ are the only perfect squares between $3$ and $17$.
So the corresponding values of $x$ are $2$, $3$, and $4$.