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

Question

Responses

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

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

(−1.6,4.2)

Answer 1

To determine which point shows a solution to the simultaneous inequalities \( y > 12x + 5 \) and \( y < -2x + 1 \), we will evaluate each option.

1. For the point \((-5, 5)\):

   • \( y > 12x + 5 \):
     • \( 5 > 12(-5) + 5 \)
     • \( 5 > -60 + 5 \)
     • \( 5 > -55 \) (True)
   • \( y < -2x + 1 \):
     • \( 5 < -2(-5) + 1 \)
     • \( 5 < 10 + 1 \)
     • \( 5 < 11 \) (True)
   • Thus, \((-5, 5)\) satisfies both inequalities.

2. For the point \((4, 8)\):

   • \( y > 12x + 5 \):
     • \( 8 > 12(4) + 5 \)
     • \( 8 > 48 + 5 \)
     • \( 8 > 53 \) (False)
   • Thus, \((4, 8)\) does not satisfy both inequalities.

3. For the point \((-1.6, 4.2)\):

   • \( y > 12x + 5 \):
     • \( 4.2 > 12(-1.6) + 5 \)
     • \( 4.2 > -19.2 + 5 \)
     • \( 4.2 > -14.2 \) (True)
   • \( y < -2x + 1 \):
     • \( 4.2 < -2(-1.6) + 1 \)
     • \( 4.2 < 3.2 + 1 \)
     • \( 4.2 < 4.2 \) (False)
   • Thus, \((-1.6, 4.2)\) does not satisfy both inequalities.

Based on the evaluations, the only point that satisfies both inequalities is \((-5, 5)\).

Therefore, the correct answer is:

\((-5, 5)\)

Answer 2

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 two solutions.
The system has two solutions.

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

The system has infinite solutions.
The system has infinite solutions.

The system has one solution.

Answer 3

Based on the description provided, it seems that the system of inequalities is graphed on a Cartesian coordinate plane with two dashed lines, which typically represent boundary lines for the inequalities.

1. Dashed lines imply that the points on the lines are not included in the solutions (i.e., the inequalities are strict: \( y < \) or \( y > \)).
2. The regions of the graph that are marked depend on the direction of the inequalities (above or below the lines).
3. The overlapping region between the two dashed lines (if they intersect) indicates where the solutions to the inequalities exist.

Given these characteristics, if both inequalities create a region where they overlap at multiple points, then it is plausible that there are infinitely many solutions. This overlap indicates a continuous range of points that satisfy both inequalities.

Considering that in typical scenarios like this:

• If the two regions do intersect and create a common area, then the system has infinite solutions.
• If they do not intersect at all, then there are no solutions.
• If there’s only a single point of intersection, it would indicate one solution.
• Two distinct solutions as a result of intersecting lines would be unlikely since that would usually suggest a single intersection point.

Given the nature of the graph described (with dashed lines and potentially overlapping regions), the most reasonable conclusion is:

The system has infinite solutions.

Please confirm the specific details (such as the orientation and intersections of the lines) from the graph to ensure this interpretation is correct.

Answer 4

Determine which of the graphs is the graph of a system of inequalities with no solutions.(1 point)
Responses

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

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

A Cartesian coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. A solid line with arrows at both ends is plotted, and various regions of the graph are marked.
Image with alt text: A Cartesian coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. A solid line with arrows at both ends is plotted, and various regions of the graph are marked.

Two dashed lines and four points are graphed on a coordinate plane. The x-axis ranges from negative 6 to 30 in increments of 2. The y-axis ranges from negative 5 to 15 in increments of 1.

Answer 5

To determine which of the graphs represents a system of inequalities with no solutions, we need to consider how inequalities and their corresponding graphical representations work:

1. Solid Lines: These indicate that the points on the line are included in the inequality (i.e., the inequalities are non-strict: \( \leq \) or \( \geq \)).

2. Dashed Lines: These indicate that the points on the line are not included in the inequality (i.e., the inequalities are strict: \( < \) or \( > \)).

3. No Solutions: A system of inequalities has no solutions if the regions that satisfy each inequality do not overlap at all; that is, they do not share any common area.

Given this information, let's analyze the provided graph descriptions:

1. First Graph: Two solid lines with marked regions.

2. Second Graph: Also two solid lines with marked regions.

3. Third Graph: A single solid line with marked regions.

4. Fourth Graph: Two dashed lines with four points plotted.

Analysis:

• The scenarios with two solid lines (first and second graphs) could potentially intersect or create overlapping regions, so they might have solutions.
• The single solid line (the third graph) may still provide a solution depending on the marked regions.
• The fourth graph with two dashed lines and four points does not typically provide insight into solutions by itself without knowing the inequalities represented and their relationships.

For a system of inequalities with no solutions, we’re particularly looking for a situation where:

• Both solid lines may indicate specific non-overlapping regions, which would be the most promising candidates for having no solution.

Conclusion: Between the options provided:

• The most likely candidates for having no solutions are the first two graphs with two solid lines. If those two solid lines represent inequalities that do not overlap (e.g., one is \( y \leq mx + b \) and the other \( y \geq mx + c \) where \( b < c \)), then they create distinct regions that do not intersect.

Without the specific images or further context on the corresponding inequalities, a definitive answer can’t be provided, but the key takeaway is that you should look for graphs where solid lines create separate regions with no common area.

If you need further clarification based on specific graph arrangements or regions indicated, please provide that context for deeper analysis!