To solve this problem, we will carefully follow the steps below:

Problem:

We are given: $g(x) = -\frac{1}{3}|2 - 7x| + 4$ and need to evaluate: $-3[g(2)] - 10$

Step 1: Substitute $x = 2$ into $g(x)$

The function is: $g(x) = -\frac{1}{3}|2 - 7x| + 4$ Substituting $x = 2$: $g(2) = -\frac{1}{3}|2 - 7(2)| + 4$ Simplify $2 - 7(2)$: $2 - 14 = -12$ Thus: $|2 - 7(2)| = |-12| = 12$

Now substitute the absolute value: $g(2) = -\frac{1}{3}(12) + 4$ Simplify: $g(2) = -4 + 4 = 0$

Step 2: Compute $-3[g(2)] - 10$

Substitute $g(2) = 0$ into the given expression: $-3[g(2)] - 10 = -3(0) - 10$ Simplify: $-3(0) - 10 = -10$

Final Answer:

$-10$

Let me know if you have any additional questions or would like further clarification!

Here are 5 follow-up questions to expand your understanding:

1. What happens to the absolute value function when the argument is negative?
2. How does the constant $-\frac{1}{3}$ affect the output of $g(x)$?
3. Why does substituting $x = 2$ simplify this function so easily?
4. What are other methods to handle expressions with absolute values efficiently?
5. How would the solution change if the argument $|2 - 7x|$ were positive instead of negative?

Tip: When dealing with absolute values, carefully resolve the inner expression first to determine its sign, as this step is critical to ensuring the rest of the calculation is correct.