To simplify the expression [tex]\( 111 \frac{1}{5} x - \frac{7}{4} - 3 \frac{3}{10} x + 1 \frac{1}{5} \)[/tex], we will follow a detailed, step-by-step approach:
1. Convert mixed fractions to improper fractions:
- [tex]\( 111 \frac{1}{5} x \)[/tex] becomes [tex]\( \left(\frac{556}{5}\right) x \)[/tex].
- [tex]\( - \frac{7}{4} \)[/tex] stays as it is already an improper fraction.
- [tex]\( - 3 \frac{3}{10} x \)[/tex] becomes [tex]\( \left(\frac{-33}{10}\right) x \)[/tex].
- [tex]\( 1 \frac{1}{5} \)[/tex] becomes [tex]\( \frac{6}{5} \)[/tex].
2. Combine like terms:
- Combine the terms containing [tex]\( x \)[/tex]:
[tex]\[
\left(\frac{556}{5} - \frac{33}{10}\right)x
\][/tex]
- To combine these fractions, get a common denominator:
[tex]\[
\frac{556}{5} = \frac{556 \times 2}{5 \times 2} = \frac{1112}{10}
\][/tex]
Now our expression is:
[tex]\[
\left(\frac{1112}{10} - \frac{33}{10}\right)x = \frac{1112 - 33}{10}x = \frac{1079}{10}x = 107.9x
\][/tex]
- Combine the constant terms:
[tex]\[
-\frac{7}{4} + \frac{6}{5}
\][/tex]
- Find a common denominator:
[tex]\[
-\frac{7}{4} = -\frac{7 \times 5}{4 \times 5} = -\frac{35}{20}
\][/tex]
[tex]\[
\frac{6}{5} = \frac{6 \times 4}{5 \times 4} = \frac{24}{20}
\][/tex]
Now we combine them:
[tex]\[
-\frac{35}{20} + \frac{24}{20} = \frac{-35 + 24}{20} = \frac{-11}{20} = -0.55
\][/tex]
3. Combine both parts of the expression:
[tex]\[
107.9 x - 0.55
\][/tex]
Thus, the simplified expression is:
[tex]\[
107.9 x - 0.55
\][/tex]
The final simplified terms [tex]\(x\)[/tex]-term, constant term, and the overall simplified expression are:
- The coefficient of [tex]\(x\)[/tex] is [tex]\(107.9\)[/tex].
- The constant term is [tex]\(-0.55\)[/tex].
- The fully simplified expression is [tex]\(107.9 x - 0.55\)[/tex].
