Your test scores in one class are 83 and 89. What possible scores can you earn on your next test to have a test average between 86 and 90, inclusive? Your score on the third test must be between and inclusive. (Use ascending order.)
To calculate the average score, we sum all the scores and divide by the number of total tests. Let's call the score on the third test $x$.
The sum of the three scores is $83 + 89 + x$. The average is $\frac{83 + 89 + x}{3}$. To have an average between 86 and 90, inclusive, we need:
$$86 \leq \frac{83 + 89 + x}{3} \leq 90$$
Multiplying through by 3:
$$258 \leq 83 + 89 + x \leq 270$$
Adding $-83-89$ to all terms of the inequality:
$$86 \leq x \leq 98$$
So, the possible scores you can earn on your next test to have a test average between 86 and 90, inclusive, are in the range of $\boxed{86}$ to $\boxed{98}$.
To find the scores that will give you an average between 86 and 90 inclusive, you need to consider the total points you have earned so far and the number of tests.
Let's assume the third test score is x.
To have an average between 86 and 90 inclusive, the sum of the three scores divided by 3 must be between 86 and 90 inclusive:
(83 + 89 + x) / 3 ≥ 86
(83 + 89 + x) / 3 ≤ 90
Now, we can solve these inequalities to find the range of possible scores for the third test:
Lower bound:
(83 + 89 + x) / 3 ≥ 86
83 + 89 + x ≥ 86 * 3
172 + x ≥ 258
x ≥ 258 - 172
x ≥ 86
Upper bound:
(83 + 89 + x) / 3 ≤ 90
83 + 89 + x ≤ 90 * 3
172 + x ≤ 270
x ≤ 270 - 172
x ≤ 98
Therefore, the possible scores for the third test to have an average between 86 and 90 inclusive are between 86 and 98 inclusive.
In ascending order, the possible scores are: 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98.
To find the range of possible scores on the third test, we need to determine the minimum and maximum scores that would result in an average between 86 and 90, inclusive.
First, let's calculate the average of the first two tests:
(83 + 89) / 2 = 86
Now, let's calculate the minimum score needed on the third test to achieve an average of 86:
(83 + 89 + x) / 3 = 86
(83 + 89 + x) = 258
x = 258 - 83 - 89
x = 86
Therefore, the minimum score on the third test is 86.
Next, let's calculate the maximum score needed on the third test to achieve an average of 90:
(83 + 89 + x) / 3 = 90
(83 + 89 + x) = 270
x = 270 - 83 - 89
x = 98
Therefore, the maximum score on the third test is 98.
In summary, the possible scores on the third test to have a test average between 86 and 90, inclusive, are 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, and 98.