To solve the inequality [tex]\(\frac{2}{3} x - 5 > 7\)[/tex], follow these steps:
1. Isolate the variable [tex]\(x\)[/tex]:
Start by adding 5 to both sides of the inequality to eliminate the constant term on the left-hand side:
[tex]\[
\frac{2}{3} x - 5 + 5 > 7 + 5
\][/tex]
Simplifying this, we get:
[tex]\[
\frac{2}{3} x > 12
\][/tex]
2. Eliminate the fraction:
Multiply both sides of the inequality by [tex]\(\frac{3}{2}\)[/tex] to cancel out the fraction on the left-hand side:
[tex]\[
x > 12 \times \frac{3}{2}
\][/tex]
Calculate the multiplication:
[tex]\[
x > 18
\][/tex]
Thus, the solution to the inequality [tex]\(\frac{2}{3} x - 5 > 7\)[/tex] is:
[tex]\[
x > 18
\][/tex]
3. Evaluate the given options:
- [tex]\(x < 18\)[/tex]: This does not satisfy [tex]\(x > 18\)[/tex].
- [tex]\(x > 6\)[/tex]: This is true for values greater than 6, but not necessarily greater than 18.
- [tex]\(x > 18\)[/tex]: This matches our solution.
- [tex]\(x < 6\)[/tex]: This is not true for values greater than 18.
Therefore, the correct solution among the given choices is:
[tex]\[
x > 18
\][/tex]
