Which one of the following is a representation (not to scale and in bold) of all values of \( x \) satisfying the inequality \( 2 - 5x \leq \frac{-6x - 5}{3} \) on the real number line?
Show Hint
To solve inequalities involving fractions, first eliminate the fraction by multiplying both sides by the denominator, and then proceed with the algebraic steps.
First, let's solve the inequality:
\[
2 - 5x \leq \frac{-6x - 5}{3}.
\]
Multiply both sides by 3 to eliminate the denominator:
\[
3(2 - 5x) \leq -6x - 5.
\]
Expanding both sides:
\[
6 - 15x \leq -6x - 5.
\]
Now, move the terms involving \( x \) to one side:
\[
6 + 5 \leq -6x + 15x.
\]
Simplifying:
\[
11 \leq 9x.
\]
Now, divide by 9:
\[
x \geq \frac{11}{9}.
\]
Thus, the solution is \( x \geq \frac{11}{9} \), which corresponds to a closed circle on \( \frac{11}{9} \) and extending to the right.
The representation that matches this solution is option (C).
