Determine if the point (3,8) is a solution to the simultaneous inequalities y>x+5 and y≤6x−3. Select the response that gives the correct answer as well as the correct reasoning.(1 point)

Question

Responses

No, it is in the solution set for y≤6x−3, but on the line for y>x+5.
No, it is in the solution set for y less-than-or-equal-to 6 x minus 3 , but on the line for y greater than x plus 5 .

Yes, because both of these inequalities include points on the line as a solution.
Yes, because both of these inequalities include points on the line as a solution.

No, it is not in the solution set for either inequality.
No, it is not in the solution set for either inequality.

Yes, it is within the solution set for both inequalities.
Yes, it is within the solution set for both inequalities.

Answer 1

To determine if the point (3,8) is a solution to the simultaneous inequalities \( y > x + 5 \) and \( y \leq 6x - 3 \), we will evaluate each inequality using the coordinates of the point.

1. For the first inequality \( y > x + 5 \):

   • Substitute \( x = 3 \) and \( y = 8 \): \[ 8 > 3 + 5 \] \[ 8 > 8 \] This is false because 8 is not greater than 8; it is equal.

2. For the second inequality \( y \leq 6x - 3 \):

   • Substitute \( x = 3 \) and \( y = 8 \): \[ 8 \leq 6(3) - 3 \] \[ 8 \leq 18 - 3 \] \[ 8 \leq 15 \] This is true because 8 is less than or equal to 15.

Since the point (3,8) does not satisfy the first inequality \( y > x + 5 \), it cannot be a solution to the system of inequalities.

Therefore, the correct response is: No, it is in the solution set for \( y \leq 6x - 3 \), but on the line for \( y > x + 5 \).

Answer 2

Which of the following points shows a solution to the simultaneous inequalities y>12x+5 and y<−2x+1?(1 point)
Responses

(4,8)
left parenthesis 4 comma 8 right parenthesis

(−5,5)
left parenthesis negative 5 comma 5 right parenthesis

(−1.6,4.2)
left parenthesis negative 1.6 comma 4.2 right parenthesis

(1,5.5)

Answer 3

To determine which of the given points is a solution to the simultaneous inequalities \( y > 12x + 5 \) and \( y < -2x + 1 \), we will evaluate each point against both inequalities.

1. Point (4,8):

   • For \( y > 12x + 5 \): \[ 8 > 12(4) + 5 \implies 8 > 48 + 5 \implies 8 > 53 \quad \text{(False)} \]
   • For \( y < -2x + 1 \): \[ 8 < -2(4) + 1 \implies 8 < -8 + 1 \implies 8 < -7 \quad \text{(False)} \]
   • This point is not a solution.

2. Point (-5,5):

   • For \( y > 12x + 5 \): \[ 5 > 12(-5) + 5 \implies 5 > -60 + 5 \implies 5 > -55 \quad \text{(True)} \]
   • For \( y < -2x + 1 \): \[ 5 < -2(-5) + 1 \implies 5 < 10 + 1 \implies 5 < 11 \quad \text{(True)} \]
   • This point is a solution.

3. Point (-1.6,4.2):

   • For \( y > 12x + 5 \): \[ 4.2 > 12(-1.6) + 5 \implies 4.2 > -19.2 + 5 \implies 4.2 > -14.2 \quad \text{(True)} \]
   • For \( y < -2x + 1 \): \[ 4.2 < -2(-1.6) + 1 \implies 4.2 < 3.2 + 1 \implies 4.2 < 4.2 \quad \text{(False)} \]
   • This point is not a solution.

4. Point (1,5.5):

   • For \( y > 12x + 5 \): \[ 5.5 > 12(1) + 5 \implies 5.5 > 12 + 5 \implies 5.5 > 17 \quad \text{(False)} \]
   • For \( y < -2x + 1 \): \[ 5.5 < -2(1) + 1 \implies 5.5 < -2 + 1 \implies 5.5 < -1 \quad \text{(False)} \]
   • This point is not a solution.

After evaluating all points, the one that satisfies both inequalities is: (-5,5).

Answer 4

Use the image to answer the question.

A Cartesian coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. 2 dashed lines with arrows at both ends are plotted, and various regions of the graph are marked.

Select a true statement to describe the graphed system of inequalities.

(1 point)
Responses

The system has one solution.
The system has one solution.

The system has two solutions.
The system has two solutions.

The system has no solutions.
The system has no solutions.

The system has infinite solutions.

Answer 5

To accurately determine the nature of the solution set for the system of inequalities represented by the dashed lines on the Cartesian coordinate plane, we would need to consider how the lines divide the plane and what regions are shaded.

Possible Scenarios:

1. One Solution: This occurs if the lines intersect at exactly one point, and there's only one point that satisfies both inequalities.
2. Two Solutions: This might happen if there are exactly two distinct points that satisfy both inequalities, which is uncommon with linear inequalities.
3. No Solutions: This scenario arises if the lines are parallel or do not overlap at all, indicating that there are no points that satisfy both inequalities.
4. Infinite Solutions: This is the case if the shaded regions overlap, indicating that there are numerous points satisfying the inequalities.

Given the descriptions of the options and without seeing the specific image, if the dashed lines correspond to two intersecting lines and there are shaded regions indicating a common area, then the likely answer would be:

The system has infinite solutions.

However, if the dashed lines are parallel or non-overlapping, then:

The system has no solutions.

Since I cannot see the image, I recommend you evaluate the lines' positions and their shaded regions to determine which option is correct based on the scenarios outlined. If the inequality regions overlap significantly, it's likely to have infinite solutions. If the lines intersect in a confined way, it might represent infinite solutions as well, but if they don't overlap, it would have no solutions.