Graph the system of linear inequalities and determine if the point (−4,−6)

Question

(
−
4
,
−
6
)
is a solution to the system.

y≥14x−5
𝑦
≥
1
4
𝑥
−
5

y<−2x+5
𝑦
<
−
2
𝑥
+
5

(1 point)
Responses

No, the point (−4,−6)
(
−
4
,
−
6
)
lies on the border of the overlapping shaded region on the solid line.
No, the point left parenthesis negative 4 comma negative 6 right parenthesis lies on the border of the overlapping shaded region on the solid line.

No, the point is not located within the overlapping shaded region.
No, the point is not located within the overlapping shaded region.

Yes, the point (−4,−6)
(
−
4
,
−
6
)
lies on the border of the overlapping shaded region on the solid line.
Yes, the point left parenthesis negative 4 comma negative 6 right parenthesis lies on the border of the overlapping shaded region on the solid line.

Yes, the point (−4,−6)
(
−
4
,
−
6
)
lies on the border of the overlapping shaded region

Answer 1

To determine if the point \((-4, -6)\) is a solution to the given system of linear inequalities, we first need to analyze each inequality and see if the point satisfies them.

1. For the inequality \(y \geq \frac{1}{4}x - 5\):

   • Substitute \(x = -4\) and \(y = -6\): \[ -6 \geq \frac{1}{4}(-4) - 5 \] \[ -6 \geq -1 - 5 \] \[ -6 \geq -6 \] This inequality is true (the point lies on the line).

2. For the inequality \(y < -2x + 5\):

   • Substitute \(x = -4\) and \(y = -6\): \[ -6 < -2(-4) + 5 \] \[ -6 < 8 + 5 \] \[ -6 < 13 \] This inequality is also true (the point is below the line).

Since both inequalities are true and \((-4, -6)\) satisfies the first inequality (as it lies on the line) and is below the second line, we conclude that the point \((-4, -6)\) is within the shaded region defined by the two inequalities.

Now, examining the responses:

• The point lies on the border of the first inequality and below the second.
• Therefore, the most appropriate response would be:

Yes, the point (-4, -6) lies on the border of the overlapping shaded region on the solid line.