To solve the problem, we need to find the value of [tex]\( x \)[/tex] that makes the mean of the given expressions equal to [tex]\(\frac{43}{4}\)[/tex].
Given expressions:
[tex]\[
\begin{array}{cccc}
x-1 & 2x-3 & x+4 & 5x-4
\end{array}
\][/tex]
1. Set Up the Mean Calculation:
The mean of these expressions is calculated as:
[tex]\[
\text{Mean} = \frac{(x-1) + (2x-3) + (x+4) + (5x-4)}{4}
\][/tex]
2. Sum the Expressions:
Combine all terms:
[tex]\[
(x-1) + (2x-3) + (x+4) + (5x-4)
\][/tex]
3. Simplify the Sum:
Combine like terms:
[tex]\[
x + 2x + x + 5x - 1 - 3 + 4 - 4
\][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[
9x - 4
\][/tex]
4. Set up the Equation with the Given Mean:
The mean is given as [tex]\(\frac{43}{4}\)[/tex]. Now we set up the equation:
[tex]\[
\frac{9x - 4}{4} = \frac{43}{4}
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Multiply both sides by 4 to clear the denominator:
[tex]\[
9x - 4 = 43
\][/tex]
Add 4 to both sides:
[tex]\[
9x = 47
\][/tex]
Divide by 9:
[tex]\[
x = \frac{47}{9}
\][/tex]
The value of [tex]\( x \)[/tex] is approximately [tex]\( 5.222 \)[/tex].
6. Evaluate Each Expression:
Substitute [tex]\( x = \frac{47}{9} \)[/tex] into each expression:
[tex]\[
x - 1 = \frac{47}{9} - 1 = \frac{47}{9} - \frac{9}{9} = \frac{38}{9} \approx 4.222
\][/tex]
[tex]\[
2x - 3 = 2 \left( \frac{47}{9} \right) - 3 = \frac{94}{9} - \frac{27}{9} = \frac{67}{9} \approx 7.444
\][/tex]
[tex]\[
x + 4 = \frac{47}{9} + 4 = \frac{47}{9} + \frac{36}{9} = \frac{83}{9} \approx 9.222
\][/tex]
[tex]\[
5x - 4 = 5 \left( \frac{47}{9} \right) - 4 = \frac{235}{9} - \frac{36}{9} = \frac{199}{9} \approx 22.111
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\(\frac{47}{9} \approx 5.222\)[/tex], and the evaluated expressions at this [tex]\( x \)[/tex] value are approximately:
1. [tex]\( x-1 \approx 4.222 \)[/tex]
2. [tex]\( 2x-3 \approx 7.444 \)[/tex]
3. [tex]\( x+4 \approx 9.222 \)[/tex]
4. [tex]\( 5x-4 \approx 22.111 \)[/tex]
These values fulfill the given condition that the mean is [tex]\(\frac{43}{4}\)[/tex].
