Mark's final grade is determined by the average of four equally weighted 100-point tests. To get a C for the semester, Mark's average score must be between 70 and 79, inclusive.

Let $x$ represent Mark's score on the fourth test.

Step 1: Set up the average equation

Mark's average score for the four tests will be: $\text{Average} = \frac{76 + 94 + 77 + x}{4}$ For Mark to earn a C, the average must satisfy the inequality: $70 \leq \frac{76 + 94 + 77 + x}{4} \leq 79$

Step 2: Solve the inequality

First, simplify the sum of the three scores: $76 + 94 + 77 = 247$ So the inequality becomes: $70 \leq \frac{247 + x}{4} \leq 79$ Now, multiply all parts of the inequality by 4 to eliminate the denominator: $70 \times 4 \leq 247 + x \leq 79 \times 4$ $280 \leq 247 + x \leq 316$

Step 3: Isolate $x$

Subtract 247 from all parts of the inequality: $280 - 247 \leq x \leq 316 - 247$ $33 \leq x \leq 69$

Step 4: Conclusion

Mark's score on the fourth test must be between 33 and 69, inclusive, for him to earn a C for the semester. Therefore, the range of scores is:

$x \in [33, 69]$

Would you like more details or have any questions?

Here are 5 related questions you might consider:

1. What happens if Mark scores below 33 on the fourth test?
2. What score does Mark need on the fourth test to get an average of exactly 75?
3. If the weight of the tests were not equal, how would this change the solution?
4. What range of scores would give Mark a B (average between 80 and 89)?
5. How does the number of tests affect the range for a desired average?

Tip: When solving inequalities involving averages, always check if multiplying or dividing by a positive number will simplify the equation without changing the inequality.