Question

$\

$$\begin{cases} 5y + 10x < 5 \\\\ y \\geq 3x - 4 \\end{cases}$$

$

Explanation:

Step1: Simplify first inequality

Divide all terms by 5:
$$\frac{5y}{5} + \frac{10x}{5} < \frac{5}{5}$$
$$y + 2x < 1$$
Rearrange to solve for $y$:
$$y < -2x + 1$$

Step2: Rewrite system clearly

The simplified system is:

$$\begin{cases} y < -2x + 1 \\ y \geq 3x - 4 \end{cases}$$

Step3: Find intersection point

Set the two expressions equal to find where the lines cross:
$$3x - 4 = -2x + 1$$
Add $2x$ to both sides:
$$5x - 4 = 1$$
Add 4 to both sides:
$$5x = 5$$
Divide by 5:
$$x = 1$$
Substitute $x=1$ into $y=3x-4$:
$$y = 3(1) - 4 = -1$$
Intersection point: $(1, -1)$

Answer:

The simplified system of inequalities is $\boldsymbol{

$$\begin{cases} y < -2x + 1 \\ y \geq 3x - 4 \end{cases}$$

}$, and the boundary lines intersect at the point $\boldsymbol{(1, -1)}$. The solution set is the region where $y$ is above/on the line $y=3x-4$ and below the line $y=-2x+1$.