Question

Which point is a solution to the simultaneous inequalities y<1/2x and y>1/3x+3 ? (1 point) Responses (3,6) left parenthesis 3 comma 6 right parenthesis (10,5) left parenthesis 10 comma 5 right parenthesis (25,12) left parenthesis 25 comma 12 right parenthesis (2,2) left parenthesis 2 comma 2 right parenthesis Skip to navigation

Answer 1

To determine which point is a solution to the simultaneous inequalities \( y < \frac{1}{2}x \) and \( y > \frac{1}{3}x + 3 \), we will test each of the provided points:

1. Point (3, 6):

   • \( y < \frac{1}{2}x \): \( 6 < \frac{1}{2} \cdot 3 = 1.5 \) ➔ False
   • \( y > \frac{1}{3}x + 3 \): \( 6 > \frac{1}{3} \cdot 3 + 3 = 4 \) ➔ True
   • Overall: False since the first inequality is not satisfied.

2. Point (10, 5):

   • \( y < \frac{1}{2}x \): \( 5 < \frac{1}{2} \cdot 10 = 5 \) ➔ False
   • \( y > \frac{1}{3}x + 3 \): \( 5 > \frac{1}{3} \cdot 10 + 3 \approx 6.33 \) ➔ False
   • Overall: False since both inequalities are not satisfied.

3. Point (25, 12):

   • \( y < \frac{1}{2}x \): \( 12 < \frac{1}{2} \cdot 25 = 12.5 \) ➔ True
   • \( y > \frac{1}{3}x + 3 \): \( 12 > \frac{1}{3} \cdot 25 + 3 \approx 11.33 \) ➔ True
   • Overall: True since both inequalities are satisfied.

4. Point (2, 2):

   • \( y < \frac{1}{2}x \): \( 2 < \frac{1}{2} \cdot 2 = 1 \) ➔ False
   • \( y > \frac{1}{3}x + 3 \): \( 2 > \frac{1}{3} \cdot 2 + 3 \approx 4 \) ➔ False
   • Overall: False since both inequalities are not satisfied.

From this analysis, the only point that satisfies both inequalities is (25, 12).

Answer 2

Which of the following points shows a solution to the simultaneous inequalities y>1/2x+5 and y<−2x+1 ?(1 point) Responses (−1.6,4.2) left parenthesis negative 1.6 comma 4.2 right parenthesis (4,8) left parenthesis 4 comma 8 right parenthesis (−5,5) left parenthesis negative 5 comma 5 right parenthesis (1,5.5)

Answer 3

To determine which point is a solution to the simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \), we will test each of the provided points:

1. Point \((-1.6, 4.2)\):

   • For \( y > \frac{1}{2}x + 5 \): \[ 4.2 > \frac{1}{2} \cdot (-1.6) + 5 \implies 4.2 > -0.8 + 5 \implies 4.2 > 4.2 \quad \text{(False)} \]
   • For \( y < -2x + 1 \): \[ 4.2 < -2 \cdot (-1.6) + 1 \implies 4.2 < 3.2 + 1 \implies 4.2 < 4.2 \quad \text{(False)} \]
   • Overall: False since both inequalities are not satisfied.

2. Point \((4, 8)\):

   • For \( y > \frac{1}{2}x + 5 \): \[ 8 > \frac{1}{2} \cdot 4 + 5 \implies 8 > 2 + 5 \implies 8 > 7 \quad \text{(True)} \]
   • For \( y < -2x + 1 \): \[ 8 < -2 \cdot 4 + 1 \implies 8 < -8 + 1 \implies 8 < -7 \quad \text{(False)} \]
   • Overall: False since the second inequality is not satisfied.

3. Point \((-5, 5)\):

   • For \( y > \frac{1}{2}x + 5 \): \[ 5 > \frac{1}{2} \cdot (-5) + 5 \implies 5 > -2.5 + 5 \implies 5 > 2.5 \quad \text{(True)} \]
   • For \( y < -2x + 1 \): \[ 5 < -2 \cdot (-5) + 1 \implies 5 < 10 + 1 \implies 5 < 11 \quad \text{(True)} \]
   • Overall: True since both inequalities are satisfied.

4. Point \((1, 5.5)\):

   • For \( y > \frac{1}{2}x + 5 \): \[ 5.5 > \frac{1}{2} \cdot 1 + 5 \implies 5.5 > 0.5 + 5 \implies 5.5 > 5.5 \quad \text{(False)} \]
   • For \( y < -2x + 1 \): \[ 5.5 < -2 \cdot 1 + 1 \implies 5.5 < -2 + 1 \implies 5.5 < -1 \quad \text{(False)} \]
   • Overall: False since both inequalities are not satisfied.

From this analysis, the only point that satisfies both inequalities is (-5, 5).

Answer 4

Use the image to answer the question. 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 no solutions. The system has no solutions. The system has infinite solutions. The system has infinite solutions. The system has two solutions. The system has two solutions. Skip to navigation

Answer 5

Without access to the specific image you are referencing, I can't directly evaluate the graph of the system of inequalities. However, I can provide you with guidelines on how to determine which statement is true:

1. One Solution: This occurs when the lines (or curves) representing the inequalities intersect exactly at one point, and that point satisfies both inequalities.

2. No Solutions: This occurs when the areas defined by the inequalities do not overlap at all. For instance, this is true if one line's area is entirely above the other while the other is entirely below or to the side.

3. Infinite Solutions: This occurs when the lines overlap or are the same line (infinite solutions along that line), or if the area satisfying both inequalities is a continuous region that has more than one dimension (like a shaded area).

4. Two Solutions: This is generally not a possible outcome for systems of inequalities. Solutions can be individual points, none, or a continuous set (one or infinite).

If you can provide a description of the graph or the relative positions of the regions defined by the inequalities, I'd be able to assist you better in identifying the correct statement. Otherwise, please follow the guidelines above based on the image you are viewing.

Answer 6

Determine which of the graphs is the graph of a system of inequalities with no solutions.(1 point) Responses Image with alt text: 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. 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. 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. 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. Skip to navigation

Answer 7

To determine which graph represents a system of inequalities with no solutions, we need to identify a situation where the shaded regions defined by the inequalities do not overlap at all.

Here is how to analyze each option based on common characteristics of inequalities:

1. Graph with Two Dashed Lines and Four Points:

   • If the dashed lines indicate the boundaries of the inequalities and do not overlap, this could indicate that there is no solution if both inequalities define regions that entirely avoid each other.

2. Graph with One Solid Line:

   • A single line usually represents one inequality. The graph could still have solutions unless the inequality region does not overlap with any other inequalities shown.

3. Graphs with Two Solid Lines:

   • If both lines are solid and indicate that the regions defined by these inequalities overlap at a point or area, then there could be solutions.

Conclusion: From the provided descriptions, the option with two dashed lines (if that is indeed the case in the first description) would likely represent a system of inequalities with no solutions, assuming the regions indicated by the inequalities do not overlap.

Look for the graph where the dashed lines form distinct boundaries with no overlapping shaded areas representing the potential solutions, and select that one.