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

When you multiply a number by its reciprocal, you always get 1! Think of it as canceling: $\frac{b}{1} \cdot \frac{1}{b} = \frac{b \cdot 1}{1 \cdot b} = \frac{b}{b} = 1$.

Use the pattern First² - 2(First)(Last) + Last². For $(3b-4)^2$: First=3b, Last=4, so you get $(3b)^2 - 2(3b)(4) + 4^2$.

While testing with specific values can help check your work, it's not sufficient for proving the inequality! You need to algebraically expand both expressions to compare them for all positive values of b.

The comparison still holds! Since $b > 0$, the term $-24b$ is always negative, making the first expression smaller regardless of b's specific value.

After expanding, the expressions differ only by $-24b$. Since $b > 0$, this term is always negative, making $9b^2 - 24b + 16 < 9b^2 + 16$.