Answer:
Explanation:
Let's call the four integers x, y, z, and w. We know that the mean of these four integers is 8, so:
x + y + z + w = 8 * 4 = 32
We also know that the median of these four integers is 6, which means that two of the integers are less than 6 and two are greater than 6. The mode of 6 means that one of the integers is 6.
Let's assume that x and y are the two integers that are less than 6. This means that z and w are the two integers that are greater than 6. We can then set up the following system of equations to represent this information:
x + y + z + w = 32
x + y < 12
z + w > 12
We also know that the range of these four integers is 14, which is the difference between the largest and smallest integers. Since z and w are the two largest integers, the range is equal to w - x. We can set up another equation to represent this information:
w - x = 14
We can solve this system of equations using substitution. First, we can solve the second equation for x:
x = 12 - y
We can then substitute this expression for x in the third equation to get:
w - (12 - y) = 14
w = 14 + y
Finally, we can substitute this expression for w in the first equation to get:
x + y + z + (14 + y) = 32
x + 2y + z = 18
We can then substitute the expression for x in the third equation to get:
(12 - y) + 2y + z = 18
-y + 2y + z = 18
z = 18 - y
We can then substitute this expression for z in the third equation to get:
w = 14 + y = 14 + (18 - z) = 14 + 18 - y = 32 - y
Substituting this expression for w in the first equation gives us:
x + y + z + (32 - y) = 32
x + 2y + z = 32
We can then substitute the expressions for x and z in the third equation to get:
(12 - y) + 2y + (18 - y) = 32
-y + 2y + 18 - y = 32
18 = 32
This equation is clearly not true, so the original assumptions we made about which integers were less than 6 and which were greater than 6 must be incorrect.
However, we can use a similar process to solve for the other possibility: that x and z are the two integers that are less than 6, and y and w are the two integers that are greater than 6.
In this case, we can set up the following system of equations:
x + y + z + w = 32
x + z < 12
y + w > 12
w - x = 14
Solving this system of equations using substitution gives us the solution x = 4, y = 10, z = 6, and w = 8.
Therefore, the four integers are 4, 6, 8, and 10.