Answer :

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]