Answer :

To solve the inequality [tex]\(2(x - 2) < \frac{x + 1}{3}\)[/tex], let's go through the solution step-by-step:

1. Expand and Simplify the Inequality:
Start by distributing the 2 on the left side of the inequality:
[tex]\[ 2(x - 2) = 2x - 4 \][/tex]
Thus, the inequality becomes:
[tex]\[ 2x - 4 < \frac{x + 1}{3} \][/tex]

2. Clear the Fraction:
To eliminate the fraction, multiply every term in the inequality by 3:
[tex]\[ 3(2x - 4) < 3 \cdot \frac{x + 1}{3} \][/tex]
Simplifying this gives:
[tex]\[ 6x - 12 < x + 1 \][/tex]

3. Isolate the Variable:
Next, we want to get all the [tex]\(x\)[/tex] terms on one side and the constants on the other. Start by subtracting [tex]\(x\)[/tex] from both sides of the inequality:
[tex]\[ 6x - x - 12 < 1 \][/tex]
Which simplifies to:
[tex]\[ 5x - 12 < 1 \][/tex]

4. Solve for [tex]\(x\)[/tex]:
Add 12 to both sides of the inequality:
[tex]\[ 5x < 1 + 12 \][/tex]
Simplifying this gives:
[tex]\[ 5x < 13 \][/tex]

5. Divide by the Coefficient of [tex]\(x\)[/tex]:
Finally, divide both sides by 5 to isolate [tex]\(x\)[/tex]:
[tex]\[ x < \frac{13}{5} \][/tex]

6. Conclusion:
Since there are no restrictions on the lower bound of [tex]\(x\)[/tex], the solution is:
[tex]\[ -\infty < x < \frac{13}{5} \][/tex]

In interval notation, this can be expressed as:
[tex]\[ x \in \left(-\infty, \frac{13}{5}\right) \][/tex]

Thus, the solution to the inequality [tex]\(2(x - 2) < \frac{x + 1}{3}\)[/tex] is:
[tex]\[ -\infty < x < \frac{13}{5} \][/tex]