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 ≥ 3x - 1

Explanation:

Step1: Graph the line $y=-x + 3$

The line $y=-x + 3$ has a $y$-intercept of 3 and a slope of - 1. Since the inequality is $y < -x+3$, we draw a dashed line and shade the region below the line.

Step2: Graph the line $y = 3x-1$

The line $y = 3x-1$ has a $y$-intercept of - 1 and a slope of 3. Since the inequality is $y\geq3x - 1$, we draw a solid line and shade the region above the line.

Step3: Find a point in the solution - set

To find the intersection point of $y=-x + 3$ and $y = 3x-1$, we set $-x + 3=3x-1$.
Adding $x$ to both sides gives $3=4x-1$.
Adding 1 to both sides: $4 = 4x$, so $x = 1$.
Substituting $x = 1$ into $y=-x + 3$ gives $y=-1 + 3=2$.
A point in the solution - set can be $(1,2)$ (since it satisfies both inequalities: $2< - 1+3$ is false, but if we consider the general process of finding a point in the overlapping shaded region, we can also test other points. Let's take $x = 0$, for $y=-x + 3$, $y = 3$ and for $y = 3x-1$, $y=-1$. A point like $(0,0)$:
For $y=-x + 3$, $0<3$ (true), for $y = 3x-1$, $0\geq - 1$ (true)). So a point in the solution set is $(0,0)$.

Answer:

$(0,0)$