Which of the following number line represents the solution of the inequation: $2 y - 3 < y + 2 \leq 3 y + 5; y \in R$ ?

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Solution

The correct option is A

Given: $2 y - 3 < y + 2 \leq 3 y + 5$
let's separate the given inequation in two inequation;

$\begin{matrix} 2 y - 3 < y + 2 2 y - y < 2 + 3 y < 5 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} y + 2 \leq 3 y + 5 - 2 y \leq 3 - y \leq \frac{3}{2} Dividing by -1 and reversing the inequality sign we get y \geq - \frac{3}{2} - \frac{3}{2} \leq y \end{matrix}$
Hence, the solution set is $= y : y ϵ R, - \frac{3}{2} \leq y < 5$

The obtained solution is represented on the number line as follows:

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