Let's simplify the given expression step-by-step:
[tex]\[
5.5(x + 6 \frac{1}{2}) - \left(x + 9 \frac{1}{3}\right) - (19 - x)
\][/tex]
First, we will simplify each term separately.
1. Simplifying [tex]\(5.5(x + 6 \frac{1}{2})\)[/tex]:
- Convert [tex]\(6 \frac{1}{2}\)[/tex] to an improper fraction: [tex]\(6 + 0.5 = 6.5\)[/tex]
- Multiply by 5.5:
[tex]\[
5.5(x + 6.5) = 5.5x + 5.5 \cdot 6.5 = 5.5x + 35.75
\][/tex]
So, [tex]\(5.5(x + 6 \frac{1}{2}) = 5.5x + 35.75\)[/tex].
2. Simplifying [tex]\(-\left(x + 9 \frac{1}{3}\right)\)[/tex]:
- Convert [tex]\(9 \frac{1}{3}\)[/tex] to an improper fraction: [tex]\(9 + \frac{1}{3} = \frac{27}{3} + \frac{1}{3} = \frac{28}{3}\)[/tex]
- Applying the negative sign inside:
[tex]\[
-\left(x + \frac{28}{3}\right) = -x - \frac{28}{3}
\][/tex]
3. Simplifying [tex]\(-(19 - x)\)[/tex]:
- Distribute the negative sign inside:
[tex]\[
-19 + x
\][/tex]
- which simplifies to:
[tex]\[
x - 19
\][/tex]
Now, let's combine all three simplified expressions:
[tex]\[
(5.5x + 35.75) + \left(-x - \frac{28}{3}\right) + (x - 19)
\][/tex]
Combining like terms:
[tex]\[
5.5x - x + x + 35.75 - 19 - \frac{28}{3}
\][/tex]
First, combine the [tex]\(x\)[/tex] terms:
[tex]\[
5.5x - x + x = 5.5x
\][/tex]
Next, combine the constant terms:
[tex]\[
35.75 - 19 - \frac{28}{3}
\][/tex]
To combine these, convert all constants to a common form. First, let's convert everything to decimals:
[tex]\[
35.75 - 19 - 9.\overline{3}
\][/tex]
Calculate:
[tex]\[
35.75 - 19 = 16.75
\][/tex]
[tex]\[
16.75 - 9.\overline{3} = 7.416666\overline{6}
\][/tex]
So, the combined expression is:
[tex]\[
5.5x + 7.416666\overline{6}
\][/tex]
Thus, the simplified expression is:
[tex]\[
5.5x + 7.41666666666666
\][/tex]
