Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.
$y < -x + 3$
$y \geq 3x - 1$

Explanation:

Step1: Identify boundary lines

For $y < -x + 3$: boundary is dashed line $y=-x+3$ (since $<$ means no equality). For $y \geq 3x - 1$: boundary is solid line $y=3x-1$ (since $\geq$ includes equality).

Step2: Find intersection of boundaries

Set $-x+3 = 3x-1$:

$$\begin{align*} 3 + 1 &= 3x + x\\ 4 &= 4x\\ x &= 1 \end{align*}$$

Substitute $x=1$ into $y=-x+3$: $y = -1 + 3 = 2$. Intersection is $(1,2)$.

Step3: Verify shaded regions

For $y < -x+3$: shade below the dashed line. For $y \geq 3x-1$: shade above the solid line. The overlapping (double-shaded) region is the solution set.

Step4: Pick a point in overlap

Choose a point in the double-shaded area, e.g., $(0,2)$:
Check $y < -x+3$: $2 < 0+3$ (true). Check $y \geq 3x-1$: $2 \geq 0-1$ (true).

Answer:

A point in the solution set is $\boldsymbol{(0,2)}$ (other valid points include $(1,2)$, $(0,0)$ etc.)