To solve the inequality [tex]\(\frac{1}{5}(2-x) + 1 < \frac{3}{4}(3x - 4)\)[/tex], we will proceed step-by-step to isolate [tex]\(x\)[/tex] and find the solution set.
1. Expand both sides of the inequality:
[tex]\[
\frac{1}{5}(2 - x) + 1 < \frac{3}{4}(3x - 4)
\][/tex]
First, we'll expand the terms inside the parentheses.
[tex]\[
\frac{1}{5} \cdot 2 - \frac{1}{5} \cdot x + 1 < \frac{3}{4} \cdot 3x - \frac{3}{4} \cdot 4
\][/tex]
This simplifies to:
[tex]\[
\frac{2}{5} - \frac{x}{5} + 1 < \frac{9x}{4} - 3
\][/tex]
2. Combine constants on the left-hand side:
Next, combine [tex]\(\frac{2}{5}\)[/tex] and [tex]\(1\)[/tex] on the left-hand side.
[tex]\[
\frac{2}{5} + 1 - \frac{x}{5} < \frac{9x}{4} - 3
\][/tex]
Converting [tex]\(1\)[/tex] to a fraction with a common denominator of 5:
[tex]\[
\frac{2}{5} + \frac{5}{5} - \frac{x}{5} < \frac{9x}{4} - 3
\][/tex]
[tex]\(\frac{2}{5} + \frac{5}{5}\)[/tex] simplifies to [tex]\(\frac{7}{5}\)[/tex]:
[tex]\[
\frac{7}{5} - \frac{x}{5} < \frac{9x}{4} - 3
\][/tex]
3. Isolate the variable [tex]\(x\)[/tex] on one side:
To isolate [tex]\(x\)[/tex], we'll move all terms involving [tex]\(x\)[/tex] to the right-hand side and constants to the left-hand side. First, subtract [tex]\(\frac{7}{5}\)[/tex] from both sides:
[tex]\[
-\frac{x}{5} < \frac{9x}{4} - 3 - \frac{7}{5}
\][/tex]
Convert [tex]\(-3\)[/tex] to a fraction with a common denominator of 5:
[tex]\[
-3 = -\frac{15}{5}
\][/tex]
Combining the constants:
[tex]\[
-\frac{x}{5} < \frac{9x}{4} - \frac{15}{5} - \frac{7}{5}
\][/tex]
Simplifying the constants:
[tex]\[
-\frac{x}{5} < \frac{9x}{4} - \frac{22}{5}
\][/tex]
4. Clear the fractions by finding a common denominator (or multiplying through by the least common multiple, which is 20):
Multiply every term by [tex]\(20\)[/tex] to clear the fractions:
[tex]\[
20 \left( -\frac{x}{5} \right) < 20 \left( \frac{9x}{4} \right) - 20 \left( \frac{22}{5} \right)
\][/tex]
Simplifying:
[tex]\[
-4x < 45x - 88
\][/tex]
5. Combine like terms:
Add [tex]\(4x\)[/tex] to both sides to bring the terms involving [tex]\(x\)[/tex] together:
[tex]\[
0 < 49x - 88
\][/tex]
Add [tex]\(88\)[/tex] to both sides to move the constant term to the left:
[tex]\[
88 < 49x
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
Finally, divide both sides by [tex]\(49\)[/tex] to isolate [tex]\(x\)[/tex]:
[tex]\[
\frac{88}{49} < x
\][/tex]
Simplifying the fraction:
[tex]\[
x > \frac{88}{49}
\][/tex]
[tex]\[
x > 1.79591836734694
\][/tex]
Putting everything together, the solution to the inequality is:
[tex]\[
x > 1.79591836734694
\][/tex]
In interval notation, it is expressed as:
[tex]\[
(1.79591836734694, \infty)
\][/tex]
