Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.
$y < -x + 3$
$y \geq 2x - 1$

Explanation:

Step1: Find intersection of lines

Set $-x+3 = 3x-1$, solve for $x$:

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

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

Step2: Verify inequality regions

• For $y < -x+3$: Shade below the dashed line $y=-x+3$.
• For $y \geq 3x-1$: Shade above the solid line $y=3x-1$.

Step3: Identify overlapping region

The solution is the overlapping shaded area, bounded by the two lines and their intersection.

Answer:

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