Discussion
Which one of the following is a representation (not to scale and in bold) of all values of \(x\) satisfying the inequality \(2 - 5x \le -\frac{6x-5}{3}\) on the real number line?
- A.
- B.
- C.
- D.
Answer: C
The solution set is shown in (C): all x from a point just to the right of 0 onwards.
Multiply both sides by 3 and collect terms:
\[ \begin{aligned} 3(2 - 5x) &\le -(6x - 5) \\ 6 - 15x &\le -6x + 5 \\ 1 &\le 9x \\ x &\ge \frac{1}{9} \end{aligned} \]
So the bold part starts at a filled dot at the small positive value 1/9 and runs to the right, which is line (C).
Line (A) shows x at most a positive value, the inequality reversed. Line (B) starts the set at a negative value, which no correct rearrangement gives, since 1/9 is positive. Line (D) shows two separate rays, the shape of a quadratic inequality, not a linear one.
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