To solve for the value of [tex]\( x \)[/tex] that will make 8 the arithmetic mean between 3 and [tex]\( 2x + 5 \)[/tex], we can proceed as follows:
1. Understand the Arithmetic Mean Formula:
The arithmetic mean of two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] is given by [tex]\(\frac{a + b}{2}\)[/tex]. In this case, the arithmetic mean is given as 8. So we set up the equation:
[tex]\[
\frac{3 + (2x + 5)}{2} = 8
\][/tex]
2. Combine Like Terms:
Simplify the expression inside the parentheses:
[tex]\[
3 + 2x + 5 = 2x + 8
\][/tex]
Now substitute this back into the equation:
[tex]\[
\frac{2x + 8}{2} = 8
\][/tex]
3. Eliminate the Denominator:
Multiply both sides of the equation by 2 to clear the fraction:
[tex]\[
2x + 8 = 16
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- Subtract 8 from both sides:
[tex]\[
2x = 8
\][/tex]
- Divide by 2:
[tex]\[
x = 4
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] that makes 8 an arithmetic mean between 3 and [tex]\( 2x + 5 \)[/tex] is [tex]\( \boxed{4} \)[/tex].
