Fill in the corresponding sign given that
x > 0
(x−4)2?x2+16
To solve this problem, we'll follow these steps:
- Step 1: Expand both expressions.
- Step 2: Subtract one expression from the other to determine the inequality.
- Step 3: Analyze and conclude based on the comparison.
Now, let's work through each step:
Step 1: Expand (x−4)2:
(x−4)2=(x−4)(x−4)=x2−2⋅4⋅x+16=x2−8x+16
Step 2: Set up the inequality (x−4)2<x2+16 and substitute the expanded form:
x2−8x+16<x2+16
Step 3: Simplify the inequality by subtracting x2 and 16 from both sides:
x2−8x+16−x2−16<0
−8x<0
Solving −8x<0 gives:
x>0
This inequality holds true for all x>0.
Therefore, the inequality (x−4)2<x2+16 is correct for x>0.
Thus, the correct symbol to fill in the blank is <.