Final answer:
To find all possible values for x in a triangle with given side lengths, use the Pythagorean theorem and inequality to determine the valid range of values. The possible values for x are all real numbers less than 21. However, it is necessary to test the smallest value, x = 21, to ensure it doesn't result in a negative side length.
Step-by-step explanation:
To find all possible values for x in a triangle with sides PQ as 7x+13, QR as 10x-2, and PR as x+27, we need to consider different scenarios. We will compare the sum of two shorter sides with the longest side (the hypotenuse) using the Pythagorean theorem. If the sum of the two shorter sides is greater than the longest side, the triangle is valid. So, we can set up the following inequality: (7x+13) + (x+27) > (10x-2).
Simplifying this inequality, we get 8x + 40 > 10x - 2. By subtracting 8x from both sides, we get 40 > 2x - 2. Adding 2 to both sides, we get 42 > 2x. Dividing by 2, we get 21 > x.
Therefore, the possible values for x are all real numbers less than 21. However, we need to test the smallest value, x = 21, to ensure it doesn't result in a negative side length. Substituting x = 21 into the side lengths, we get PQ = 150,
QR = 208, and PR = 48. Since all the side lengths are positive, x = 21 is a valid solution.