To solve the inequality [tex]\(\frac{5}{18}-\frac{x-2}{9} \leq \frac{x-4}{6}\)[/tex], follow these steps:
1. Identify a common denominator for the fractions involved.
[tex]\[
\text{The common denominator for } 18, 9, \text{ and } 6 \text{ is } 18.
\][/tex]
2. Rewrite each term with the common denominator:
[tex]\[
\frac{5}{18} - \frac{x-2}{9} \leq \frac{x-4}{6}
\][/tex]
[tex]\[
\frac{5}{18} - \frac{2(x-2)}{18} \leq \frac{3(x-4)}{18}
\][/tex]
[tex]\( \text{(multiply both the numerator and the denominator of } \frac{x-2}{9} \text{ by 2 and } \frac{x-4}{6} \text{ by 3)}\)[/tex].
3. Combine the fractions on the left-hand side:
[tex]\[
\frac{5}{18} - \frac{2(x-2)}{18} = \frac{5 - 2(x-2)}{18} = \frac{5 - 2x + 4}{18} = \frac{9 - 2x}{18}
\][/tex]
4. Combine the fractions on the right-hand side:
[tex]\[
\frac{3(x-4)}{18} = \frac{3x - 12}{18}
\][/tex]
5. Rewrite the inequality with the fractions combined:
[tex]\[
\frac{9 - 2x}{18} \leq \frac{3x - 12}{18}
\][/tex]
6. Eliminate the common denominator by multiplying both sides by 18:
[tex]\[
9 - 2x \leq 3x - 12
\][/tex]
7. Combine like terms by moving [tex]\(2x\)[/tex] to the right side and -12 to the left side:
[tex]\[
9 + 12 \leq 3x + 2x
\][/tex]
[tex]\[
21 \leq 5x
\][/tex]
8. Isolate [tex]\(x\)[/tex] by dividing both sides by 5:
[tex]\[
x \geq \frac{21}{5}
\][/tex]
So we find that:
[tex]\[
x \geq 4.2
\][/tex]
The correct solution to the inequality [tex]\(\frac{5}{18} - \frac{x-2}{9} \leq \frac{x-4}{6}\)[/tex] is [tex]\(x \geq 4.2\)[/tex].
