Q.Show the solution set for the following inequation $x + 4y \leq 0$ graphically.

The line $x + 4y = 0$ passes through the origin with slope $-\frac14$; testing $(0,-1)$ gives $-4 \le 0$ (true), so shade that side. The boundary is included (solid line).

The boundary line is $x + 4y = 0$, i.e. $y = -\dfrac{x}{4}$. It passes through the origin; two convenient points are $(0, 0)$ and $(4, -1)$ (and $(-4, 1)$).

Because the inequality is $\le$ (not strict), the line itself is part of the solution set — draw it as a solid line.

Choose a test point not on the line, say $(0, -1)$:

$x + 4y = 0 + 4(-1) = -4 \le 0 \quad \text{(true)}.$

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