Question

1. $x + y \leq 2$

$2x - y \leq 1$

Explanation:

Step1: Rewrite in slope-intercept form

For $x+y\leq2$: $y\leq -x+2$
For $2x-y\leq1$: $y\geq 2x-1$

Step2: Plot boundary lines

1. For $y=-x+2$:

• y-intercept: $(0,2)$, x-intercept: $(2,0)$. Draw a solid line.

1. For $y=2x-1$:

• y-intercept: $(0,-1)$, x-intercept: $(\frac{1}{2},0)$. Draw a solid line.

Step3: Shade solution regions

1. For $y\leq -x+2$: Shade below the line.
2. For $y\geq 2x-1$: Shade above the line.

Step4: Identify intersection point

Solve

$$\begin{cases}y=-x+2\\y=2x-1\end{cases}$$

Set $-x+2=2x-1$ → $3x=3$ → $x=1$, substitute $x=1$: $y=1$. Intersection: $(1,1)$

Answer:

The solution is the overlapping shaded region bounded by the solid lines $y=-x+2$, $y=2x-1$, and their intersection point $(1,1)$.

• The region includes all points that lie below $y=-x+2$ and above $y=2x-1$.