Compare Expressions: (3b-4)² vs 9b(b+1/b)+7 with Positive b

To solve this problem, we need to compare two expressions:

• Expression 1: $(3b-4)^2$, which can be expanded as:

$(3b-4)^2 = 9b^2 - 24b + 16$

• Expression 2: $9b\left(b+\frac{1}{b}\right) + 7$, which expands to:

$9b \cdot b + 9b \cdot \frac{1}{b} + 7 = 9b^2 + 9 + 7 = 9b^2 + 16$

Now, we compare the simplified expressions:

• Expression 1: $9b^2 - 24b + 16$
• Expression 2: $9b^2 + 16$

The only difference between these expressions is the $-24b$ term in Expression 1, which makes it smaller since $-24b$ is negative for $b > 0$.

Therefore, $(3b-4)^2$ is less than $9b(b+\frac{1}{b})+7$. The correct inequality sign is < .