a student scored 93 and 91 on her first two quizzes. write and solve a comound inequality to find the possible values for a third quiz score that would give her an average between 85 and 90, inclusive.
Let's call the third quiz score "x".
To find the average, we need to sum up the three quiz scores and divide by 3.
So, the average is (93 + 91 + x)/3.
We want the average to be between 85 and 90, inclusive. This means the average should be greater than or equal to 85 and less than or equal to 90.
So, we can write the compound inequality as:
85 ≤ (93 + 91 + x)/3 ≤ 90.
To solve this compound inequality, we can start by multiplying all parts of the inequality by 3 (to eliminate the denominator):
3 * 85 ≤ 3 * [(93 + 91 + x)/3] ≤ 3 * 90.
255 ≤ 93 + 91 + x ≤ 270.
Next, we can simplify the inequality:
255 ≤ (93 + 91 + x) ≤ 270.
To isolate the "x" term, we can subtract 93 and 91 from all parts of the inequality:
255 - 93 ≤ (93 + 91 + x) - (93 + 91) ≤ 270 - 93.
162 ≤ x ≤ 177.
Therefore, the possible values for the third quiz score that would give her an average between 85 and 90, inclusive, are any values between 162 and 177 (both inclusive).
To find the possible values for the third quiz score that would give the student an average between 85 and 90, inclusive, we need to set up a compound inequality.
Let's denote the third quiz score by "x".
First, we need to find the average of the three quiz scores. The total of the first two quiz scores is 93 + 91 = 184. To maintain an average between 85 and 90, including both endpoints, the average of the three quiz scores should be within the range [85, 90].
We can find the average by dividing the sum of the three quiz scores by 3. The compound inequality can then be written as:
(184 + x) / 3 ≥ 85 and (184 + x) / 3 ≤ 90
Simplifying the above compound inequality, we get:
95 ≤ (184 + x) / 3 ≤ 90
To remove the fraction, we multiply both sides of the compound inequality by 3:
3 * 95 ≤ 184 + x ≤ 3 * 90
285 ≤ 184 + x ≤ 270
Next, we subtract 184 from all sides of the inequality:
285 - 184 ≤ 184 + x - 184 ≤ 270 - 184
101 ≤ x ≤ 86
Therefore, the compound inequality for the possible values of the third quiz score is:
101 ≤ x ≤ 86
To find the possible values for a third quiz score, let's first calculate the average of the first two quiz scores. Then we'll set up a compound inequality with the given average range.
Step 1: Find the average of the first two quiz scores.
Add the scores and divide by 2:
(93 + 91)/2 = 92
Step 2: Set up the compound inequality.
Let x be the third quiz score. We want the average of the three scores (including x) to fall between 85 and 90, inclusive. To express this, we can set up the following compound inequality:
85 ≤ (93 + 91 + x)/3 ≤ 90
Step 3: Simplify and solve the compound inequality.
Multiply each part of the inequality by 3 to eliminate the fraction:
255 ≤ 93 + 91 + x ≤ 270
Combine like terms:
255 ≤ 184 + x ≤ 270
Subtract 184 from each part of the inequality:
71 ≤ x ≤ 86
So, the compound inequality for the possible values of the third quiz score is 71 ≤ x ≤ 86.