Which of the points on the graph is a solution for the simultaneous inequalities y>4x−2

Question

𝑦
>
4
𝑥
−
2
and y≥−13x+2
𝑦
≥
−
1
3
𝑥
+
2
? Enter the x-value followed by the y-value.

(1 point)
(,)

Answer 1

To find a point that satisfies both inequalities $y>4x-2$ and $y\ge -13x+2$, we need to analyze the two equations that represent the boundaries of the inequalities.

1. First Inequality: $y>4x-2$

   • This line has a slope of 4 and a y-intercept of -2.

2. Second Inequality: $y\ge -13x+2$

   • This line has a slope of -13 and a y-intercept of 2.

The solution to the system of inequalities will be the region that is above the line $y=4x-2$ and above (or on) the line $y=-13x+2$.

To find potential points that lie in the solution area, we can substitute some values of $x$ into both equations and check for their corresponding values of $y$. Let's check a couple of points:

1. For $x=0$:

   • From $y>4(0)-2$: $y>-2$
   • From $y\ge -13(0)+2$: $y\ge 2$
   • The point (0, 2) satisfies the second inequality but not the first, as $2>-2$.

2. For $x=1$:

   • From $y>4(1)-2$: $y>2$
   • From $y\ge -13(1)+2$: $y\ge -11$
   • The point (1, 3) satisfies both inequalities as $3>2$ and $3\ge -11$.

Thus, the point (1, 3) is a solution for the simultaneous inequalities $y>4x-2$ and $y\ge -13x+2$.

Final result:

(1, 3)