To determine which operator can be inserted between the two expressions, let's evaluate each side of the inequality step by step.
First, simplify the left side:
\[ -6 + 3 \cdot 5 \]
Calculating \(3 \cdot 5\):
\[ 3 \cdot 5 = 15 \]
So,
\[ -6 + 15 = 9 \]
Now, we have:
\[ -6 + 3 \cdot 5 = 9 \]
Next, simplify the right side:
\[ 16 - \sqrt{16} + 32 \div 8 \]
First, evaluate \(\sqrt{16}\):
\[ \sqrt{16} = 4 \]
Then, calculate \(32 \div 8\):
\[ 32 \div 8 = 4 \]
Now plug those values into the expression:
\[ 16 - 4 + 4 \]
Calculating this step by step:
- \(16 - 4 = 12\)
- \(12 + 4 = 16\)
Now we have:
\[ 16 - \sqrt{16} + 32 \div 8 = 16 \]
Now, we can compare the two results:
Left side: \(9\)
Right side: \(16\)
Now we can check the possible relationships:
- \(9 > 16\) is false.
- \(9 \leq 16\) is true.
- \(9 < 16\) is true (and also fits).
- \(9 = 16\) is false.
Since both \(9 \leq 16\) and \(9 < 16\) are true, we could use either. The simplest form is to use the less-than operator.
Thus, the best operator to insert is:
\[ < \]
So the final answer is: < (less than).