To determine the value of [tex]\( x \)[/tex] where the mean of the expressions [tex]\( x+1 \)[/tex], [tex]\( x+3 \)[/tex], and [tex]\( x+5 \)[/tex] is 4, follow these steps:
1. Understand the Mean Formula:
The mean (average) of a set of numbers is given by the sum of the numbers divided by the count of numbers. For the expressions [tex]\( x+1 \)[/tex], [tex]\( x+3 \)[/tex], and [tex]\( x+5 \)[/tex], the mean is given by:
[tex]\[
\text{Mean} = \frac{(x+1) + (x+3) + (x+5)}{3}
\][/tex]
2. Set Up the Equation:
According to the problem, the mean of [tex]\( x+1 \)[/tex], [tex]\( x+3 \)[/tex], and [tex]\( x+5 \)[/tex] is 4. Therefore, we set up the equation:
[tex]\[
\frac{(x+1) + (x+3) + (x+5)}{3} = 4
\][/tex]
3. Simplify the Numerator:
Combine like terms in the numerator:
[tex]\[
(x+1) + (x+3) + (x+5) = x + 1 + x + 3 + x + 5 = 3x + 9
\][/tex]
So the equation now becomes:
[tex]\[
\frac{3x + 9}{3} = 4
\][/tex]
4. Eliminate the Denominator:
Multiply both sides of the equation by 3 to clear the fraction:
[tex]\[
3x + 9 = 4 \times 3
\][/tex]
Simplify the right-hand side:
[tex]\[
3x + 9 = 12
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
- Subtract 9 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
3x = 12 - 9
\][/tex]
[tex]\[
3x = 3
\][/tex]
- Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{3}{3}
\][/tex]
[tex]\[
x = 1
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{1} \)[/tex].
