The expression makes sense for x≤10 and x≥−10.
The expression involves the square root of a quantity that is 10 minus the absolute value of x. For the expression to make sense, the argument inside the square root must not be negative, as the square root of a negative number is undefined in the real number system.
Considering the expression 10−∣x∣, the absolute value of x ensures that 10−∣x∣ is always non-negative, as the absolute value function makes its output non-negative. Therefore, the square root of 10−∣x∣ is defined as long as 10−∣x∣ is non-negative.
To find when 10−∣x∣ is non-negative, we consider two cases:
When x≥0: In this case, ∣x∣=x, and 10−∣x∣=10−x. For the expression to be non-negative, x must be less than or equal to 10.
When x<0: In this case, ∣x∣=−x, and 10−∣x∣=10+x. For the expression to be non-negative, x must be greater than or equal to −10.
Combining these conditions, the expression 10−∣x∣ makes sense when x≤10 and x≥−10.