Question

Solve the following inequation and graph the solution set on the number line: $2y-3<y+1\le 4y+7,\forall y\in \mathrm{ℝ}$

Answer 1

Step 1: Given inequation: $2y-3<y+1\le 4y+7$.

Consider the inequality: $2y-3<y+1$

Subtract both sides of the inequality by $y$.

$$\begin{gathered} 2y-3-y<y+1-y \\ y-3<1 \end{gathered}$$

Step 2: Add $3$ to both sides of the inequality.

$$\begin{gathered} y-3+3<1+3 \\ y<4 \end{gathered}$$

Thus, the solution set of the given inequality can be given by $\left(-\infty ,4\right)$.

Step 3: Consider the inequality: $y+1\le 4y+7$

Subtract both sides of the inequality by $y$.

$$\begin{gathered} y+1-y\le 4y+7-y \\ 1\le 3y+7 \\ 3y+7\ge 1 \end{gathered}$$

Step 4: Subtract $7$ both sides of the inequality.

$$\begin{gathered} 3y+7-7\ge 1-7 \\ 3y\ge -6 \end{gathered}$$

Step 5: Divide both sides of the inequality by $3$.

$$\begin{gathered} \frac{3y}{3}\ge -\frac{6}{3} \\ y\ge -2 \end{gathered}$$

Thus, the solution set of the given inequality can be given by $[-2,\infty )$.

Therefore, the solution set of $\left(-\infty ,4\right)$ and $[-2,\infty )$ is $[-2,4)$.

Step 6: Reoresent the solution set on the number line

The given solution set can be plotted on the number line as below,