To determine the minimum score needed on the third test, let $x$ represent the score on the third test.

The average score required is at least 75. Since the student has taken three tests, we can set up the inequality as follows:

$\frac{82 + 74 + x}{3} \geq 75$

Now, solve for $x$:

1. Multiply both sides by 3 to eliminate the denominator: $82 + 74 + x \geq 225$

2. Simplify the left side: $156 + x \geq 225$

3. Subtract 156 from both sides: $x \geq 69$

Answer

The minimum score the student needs on the third test to pass is 69.

Would you like further details, or have any questions?

Here are five related questions to consider:

1. What if the minimum average required was 80 instead of 75?
2. How would the inequality change if the student scored 60 on the second test?
3. How could we generalize this problem for a course with more than three tests?
4. What happens if we only know the minimum passing average and one test score?
5. How would the result change if bonus points were added to each test?

Tip: When solving inequalities with averages, isolating the unknown variable is key to understanding minimum requirements.