To solve the inequality [tex]\(\frac{5 - 2x}{3} \leq \frac{x}{6} - 5\)[/tex], we can begin by eliminating the fractions through multiplication. Here is a step-by-step solution:
1. Identify and Simplify the Inequality:
The given inequality is:
[tex]\[
\frac{5 - 2x}{3} \leq \frac{x}{6} - 5
\][/tex]
2. Eliminate the Fractions:
We can get rid of the denominators by multiplying all terms by 6, the least common multiple of 3 and 6:
[tex]\[
6 \left( \frac{5 - 2x}{3} \right) \leq 6 \left( \frac{x}{6} \right) - 6 \cdot 5
\][/tex]
Simplifying the expressions, we get:
[tex]\[
2(5 - 2x) \leq x - 30
\][/tex]
[tex]\[
10 - 4x \leq x - 30
\][/tex]
3. Isolate the Variable [tex]\(x\)[/tex]:
Combine like terms by moving all terms involving [tex]\(x\)[/tex] to one side and the constants to the other side:
[tex]\[
10 - 4x \leq x - 30
\][/tex]
Add [tex]\(4x\)[/tex] to both sides:
[tex]\[
10 \leq x + 4x - 30
\][/tex]
[tex]\[
10 \leq 5x - 30
\][/tex]
Add 30 to both sides:
[tex]\[
40 \leq 5x
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 5 to isolate [tex]\(x\)[/tex]:
[tex]\[
\frac{40}{5} \leq x
\][/tex]
[tex]\[
8 \leq x
\][/tex]
This can be written as:
[tex]\[
x \geq 8
\][/tex]
Therefore, the solution to the inequality [tex]\(\frac{(5 - 2x)}{3} \leq \frac{x}{6} - 5\)[/tex] is:
[tex]\(\boxed{x \geq 8}\)[/tex]
So, the correct answer is:
(B) [tex]\(x \leq 8\)[/tex]
