Final answer:
To write the expression |x - 9| + |x - 7| without absolute value symbols for x in the interval (-∞, 1), both x - 9 and x - 7 are negative, so the expression simplifies to 9 - x + 7 - x, which then simplifies to 16 - 2x.
Step-by-step explanation:
To write the expression |x − 9| + |x − 7| without the absolute value symbols for the interval (∞, 1) we need to consider the definition of the absolute value function and the specific interval given. The absolute value of a number is the non-negative value of that number without regard to its sign. Thus, if the number inside the absolute value is negative, we take its opposite (change the sign), and if it's positive, we keep it as it is.
Because we are working in the interval (∞, 1), any value of x that we choose is less than both 9 and 7. Therefore, both x − 9 and x − 7 will be negative, and we need to change the sign of each expression inside the absolute values to make them positive.
The expression thus simplifies as follows:
For all x in (
∞, 1):
|x − 9| becomes -(x − 9), which is 9 − x.
|x − 7| becomes -(x − 7), which is 7 − x.
So the expression without absolute values is:
9 − x + 7 − x
Combining like terms:
16 − 2x