The mean of the numbers 0, x + 2, 3x + 6, and 4x + 8 is 4, find the value of x.

Answer Details

The mean (average) of a set of numbers is found by adding up all the numbers and dividing the sum by the number of values.

In this problem, the numbers are:
0,
\(x + 2\),
\(3x + 6\),
\(4x + 8\).

The mean of these four numbers is given as 4.

Let's write an equation for the mean:

\[ \text{Mean} = \frac{0 + (x + 2) + (3x + 6) + (4x + 8)}{4} = 4 \]

Now, simplify the numerator by combining like terms:

• Add up all the \(x\) terms: \(x + 3x + 4x = 8x\)
• Add up all the constants: \(2 + 6 + 8 = 16\)

So the numerator becomes:

\[ 0 + x + 2 + 3x + 6 + 4x + 8 = 8x + 16 \]

The equation is now:

\[ \frac{8x + 16}{4} = 4 \]

To solve for \(x\), first clear the denominator by multiplying both sides by 4:

\[ 8x + 16 = 16 \]

Subtract 16 from both sides:

\[ 8x = 0 \]

Divide both sides by 8:

\[ x = 0 \]

Conclusion: The value of \(x\) that makes the mean of the four numbers equal to 4 is \(x = 0\). The key concept is applying the formula for the mean and carefully combining terms before solving the resulting equation.