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

Analysis

To find the value of $x$, we need to use the definition of the mean (average) of a set of numbers. The mean is calculated by finding the sum of all the numbers and dividing that sum by the total count of the numbers.

Concept

The formula for the mean is:

In this problem:

• The numbers are: $0, (x + 2), (3x + 6), (4x + 8)$
• The total count of numbers is: $4$
• The mean is given as: $4$

Step-by-Step Solution

Step 1: Set up the equation.

Step 2: Simplify the numerator by combining like terms.

• Sum of $x$ terms: $x + 3x + 4x = 8x$
• Sum of constants: $0 + 2 + 6 + 8 = 16$
• The equation becomes: $\frac{8x + 16}{4} = 4$

Step 3: Solve for $x$. Multiply both sides by 4 to clear the fraction:

Subtract 16 from both sides:

Divide by 8:

Conclusion

By substituting $x = 0$ back into the original numbers, we get $0, 2, 6, 8$. The sum is $0 + 2 + 6 + 8 = 16$, and $16 \div 4 = 4$, which confirms our result.

Exam Tip

When dealing with mean problems involving algebraic expressions, always group your "like terms" (all the $x$'s together and all the constants together) before trying to solve the equation. This prevents simple arithmetic errors that often occur during the addition phase.