On the past two tests, a student scored a 75 and 83. Write and solve a compound inequality to
find the possible values for the 3rd test score that would give him an average between 85 and 90,
inclusive.
Please give me a step by step explanation so I can learn how to solve other questions like this. Thank you have a great day :)
85 </= (75 + 83 + x)/3 </= 90
85 </= (158 + x)/3 </= 90
so lower limit is
85*3 = 158+x
x = 97
upper is
90*3 = 158 + x
x = 112
so
97 </= x </= 112
if greater than 100 is not possible, stop at 100
To find the possible values for the 3rd test score, we need to set up a compound inequality. Let's break down the problem step-by-step:
Step 1: Identify the given information.
The student scored a 75 and 83 on the first two tests. We also know that the average of the three test scores should be between 85 and 90, inclusive.
Step 2: Set up the compound inequality.
Let's represent the third test score with the variable "x". The average of the three test scores is given by the formula: (75 + 83 + x) / 3. We need to ensure that this average falls between 85 and 90, inclusive.
So the compound inequality we need to solve is:
85 ≤ (75 + 83 + x) / 3 ≤ 90
Step 3: Simplify the compound inequality.
To begin simplifying, let's multiply all parts of the compound inequality by 3, as it will clear the fraction:
85 * 3 ≤ 75 + 83 + x ≤ 90 * 3
This becomes:
255 ≤ 158 + x ≤ 270
Step 4: Simplify further.
To isolate the variable "x", we need to get rid of the constants on the sides. Let's start by subtracting 158 from all parts:
255 - 158 ≤ 158 + x - 158 ≤ 270 - 158
This becomes:
97 ≤ x ≤ 112
Step 5: Interpret the solution.
The possible values for the third test score, represented by "x", are between 97 and 112, inclusive. This means the student needs to score between 97 and 112 to have an average between 85 and 90, inclusive.
So, the compound inequality to find the possible values for the third test score is 97 ≤ x ≤ 112.
To find the possible values for the 3rd test score that would give the student an average between 85 and 90, inclusive, we need to set up a compound inequality.
Step 1: Let's assign a variable to the 3rd test score. Let's call it "x".
Step 2: The average of three test scores is found by adding up all the scores and dividing by the total number of scores. In this case, we have 3 test scores.
Step 3: The average is between 85 and 90, inclusive. So, we need to set up a compound inequality to find the range of possible values for "x".
Step 4: The compound inequality for finding the average is:
(75 + 83 + x) / 3 ≥ 85 and (75 + 83 + x) / 3 ≤ 90
Step 5: Simplify the inequalities:
158 + x ≥ 255 and 158 + x ≤ 270
Step 6: Subtract 158 from both sides of both inequalities:
x ≥ 97 and x ≤ 112
Step 7: So, the range of possible values for the 3rd test score, "x", is between 97 and 112, inclusive.
Thus, the compound inequality is:
97 ≤ x ≤ 112