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]
