Which is the graph of 2x – 4y > 6? 1. On a coordinate plane, a dashed straight line with a positive slope goes through (negative 2, negative 2) and (1, 0). Everything to the right of the line is shaded. 2. On a coordinate plane, a dashed straight line with positive slope goes through (negative 3, negative 3) and (3, 0). Everything to the left of the line is shaded. 3. On a coordinate plane, a dashed straight line with a positive slope goes through (negative 2, negative 2) and (1, 0). Everything to the left of the line is shaded. 4. On a coordinate plane, a dashed straight line with positive slope goes through (negative 3, negative 3) and (3, 0). Everything to the right of the line is shaded.

Answer 1

To determine which graph corresponds to the inequality \(2x - 4y > 6\), we first need to rewrite the inequality in slope-intercept form (i.e., \(y < mx + b\) or \(y > mx + b\)).

Starting from the given inequality:

\[ 2x - 4y > 6 \]

We can solve for \(y\):

1. Subtract \(2x\) from both sides:

   \[ -4y > -2x + 6 \]

2. Divide each term by \(-4\), remembering to flip the inequality sign:

   \[ y < \frac{1}{2}x - \frac{3}{2} \]

Now, we have the inequality in the form \(y < mx + b\), where the slope \(m\) is \(\frac{1}{2}\) and the y-intercept \(b\) is \(-\frac{3}{2}\).

Next, we will graph the line \(y = \frac{1}{2}x - \frac{3}{2}\):

1. Find the y-intercept, which is \(-\frac{3}{2}\) (or \(-1.5\)). This means the line crosses the y-axis at \( (0, -1.5) \).
2. The slope \(\frac{1}{2}\) means that for every 2 units you move to the right (positive x-direction), you move 1 unit up (positive y-direction).

Using this information, we can find a couple of points through which the line passes:

• At \(x = -2\), \(y = \frac{1}{2}(-2) - \frac{3}{2} = -1 - \frac{3}{2} = -2\), so one point is \((-2, -2)\).
• At \(x = 1\), \(y = \frac{1}{2}(1) - \frac{3}{2} = \frac{1}{2} - \frac{3}{2} = -1\), yielding the point \((1, -1)\).

The line is dashed because the inequality does not include equality (\(>\) instead of \(\ge\)).

Since the inequality is \(y < \frac{1}{2}x - \frac{3}{2}\), the region that is shaded will be below the line.

Now we can analyze the given options:

1. A dashed line with a positive slope goes through \((-2, -2)\) and \((1, 0)\). Everything to the right is shaded. (Not correct - wrong shading)
2. A dashed line with a positive slope goes through \((-3, -3)\) and \((3, 0)\). Everything to the left is shaded. (Not correct - wrong line)
3. A dashed line with a positive slope goes through \((-2, -2)\) and \((1, 0)\). Everything to the left is shaded. (Not correct - wrong shading)
4. A dashed line with a positive slope goes through \((-3, -3)\) and \((3, 0)\). Everything to the right is shaded. (Not correct - wrong line)

The correct graph is not listed among the options provided since all provided graphs seem to have errors in either the line or the shading. The correct graph would have a dashed line with positive slope cutting through points corresponding to the y-intercept of \(-\frac{3}{2}\) and something that follows the slope of \(\frac{1}{2}\), with shading below the line.

Answer 2

A dashed line with a positive slope goes through (-3,-3)
and (1,0). Everything to the right is shaded.

On a coordinate plane, a dashed straight line with negative slope goes through (0, 1) and (2, negative 2). Everything to the left of the line is shaded.Which inequality is represented by the graph?y > Negative two-thirdsx + 1y < Negative two-thirdsx + 1y < Negative three-halvesx + 1y > Negative three-halvesx + 1

Answer 3

To determine which inequality is represented by each graph, let's analyze them one by one.

1. Dashed Line with Positive Slope through Points (-3, -3) and (1, 0)

Step 1: Calculate the slope. The slope \(m\) between the points \((-3, -3)\) and \((1, 0)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-3)}{1 - (-3)} = \frac{3}{4} \]

Step 2: Find the y-intercept. Using the point-slope form \(y - y_1 = m(x - x_1)\), we use the point \((1, 0)\): \[ y - 0 = \frac{3}{4}(x - 1) \] This simplifies to: \[ y = \frac{3}{4}x - \frac{3}{4} \] The line equation is: \[ y = \frac{3}{4}x - \frac{3}{4} \]

Step 3: Determine the inequality. Since the graph is shaded to the right of the line and the line is dashed, the inequality represented will be: \[ y > \frac{3}{4}x - \frac{3}{4} \]

This appears not to match any listed options. Let's move on to the second graph.

2. Dashed Line with Negative Slope through Points (0, 1) and (2, -2)

Step 1: Calculate the slope. The slope \(m\) between \((0, 1)\) and \((2, -2)\) is: \[ m = \frac{-2 - 1}{2 - 0} = \frac{-3}{2} \]

Step 2: Find the y-intercept. Using the point-slope form with the point \((0, 1)\): \[ y - 1 = -\frac{3}{2}(x - 0) \] This simplifies to: \[ y = -\frac{3}{2}x + 1 \]

Step 3: Determine the inequality. Since the graph is shaded to the left of the line and the line is dashed, the inequality represented will be: \[ y < -\frac{3}{2}x + 1 \]

Conclusion on Provided Options

From the two derived equations, we can identify that the second graph corresponds to the inequalities given. Particularly, it matches: \[ y < -\frac{3}{2}x + 1 \]

This corresponds to the inequality option:

• \(y < -\frac{3}{2}x + 1\)

Thus, the answer to the inequality represented by the second graph is: Option: \(y < -\frac{3}{2}x + 1\)

Answer 4

On a coordinate plane, a dashed straight line with positive slope goes through (0, 0.75) and (2, 4.75). Everything to the left of the line is shaded.Which inequality is represented by the graph?___ Which point is a solution to the inequality?___

Answer 5

To analyze the graph of the line and determine the inequality represented, let's follow these steps:

Step 1: Find the Equation of the Line

1. Calculate the slope (m) between the points \((0, 0.75)\) and \((2, 4.75)\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4.75 - 0.75}{2 - 0} = \frac{4}{2} = 2 \]

2. Find the y-intercept (b). Since one of the points is \((0, 0.75)\), the y-intercept \(b\) is \(0.75\).

3. Write the equation of the line in slope-intercept form \(y = mx + b\): \[ y = 2x + 0.75 \]

Step 2: Determine the Inequality

Since the graph is shaded to the left of the line and the line is dashed, the corresponding inequality is:

• \(y < 2x + 0.75\)

Step 3: Find a Point That Is a Solution to the Inequality

To find a point that satisfies the inequality \(y < 2x + 0.75\), we can test specific points either below the line or on the x-y plane.

For simplicity, we can try the origin \((0, 0)\):

1. Plug \(x = 0\) into the inequality: \[ y < 2(0) + 0.75 \implies y < 0.75 \] Since \(0 < 0.75\), the point \((0, 0)\) is a solution.

Summary

Inequality represented by the graph: \[ y < 2x + 0.75 \]

Point that is a solution to the inequality: \[ (0, 0) \] or any point that lies to the left of the line such as \((-1, 0.5)\), \((-1, 0)\), etc.

Answer 6

Which inequality is represented by the graph?
y>2x+(3/4)
y<2x+(3/4)
y<1/2x+(3/4)
y>1/2x+(3/4)



Which point is a solution to the inequality?
(-2,-4)
(-1,-2)
(1,2)
(2,5)

Answer 7

Let's analyze the graph of the line given by the inequality and determine its characteristics step by step.

Step 1: Given Information

The line has a positive slope and passes through the y-intercept, which is \((0, \frac{3}{4})\). The slope is \(2\) since the options include \(y > 2x + \frac{3}{4}\) and \(y < 2x + \frac{3}{4}\).

Step 2: Determine the Inequality

• If the line is dashed and the region above the line is shaded, the inequality would be: \[ y > 2x + \frac{3}{4} \]

• If the dashed line has shading below (left side) of the line, the inequality would be: \[ y < 2x + \frac{3}{4} \]

From the context provided about "--everything to the left" should mean the graph is likely for:

• The inequalities do not include the less than or equal (\(≤\)) signs, which means we are dealing with strict inequalities.

Step 3: Solution Point

Next, we need to determine which of the given points satisfies the inequality:

Possible Points:

1. Point (-2, -4): \[ -4 ; ? < ; 2(-2) + \frac{3}{4} \implies -4 < -4 + \frac{3}{4} \implies -4 < -3.25 ; \text{(True)} \]

2. Point (-1, -2): \[ -2 ; ? < ; 2(-1) + \frac{3}{4} \implies -2 < -2 + \frac{3}{4} \implies -2 < -1.25 ; \text{(True)} \]

3. Point (1, 2): \[ 2 ; ? < ; 2(1) + \frac{3}{4} \implies 2 < 2 + \frac{3}{4} \implies 2 < 2.75 ; \text{(True)} \]

4. Point (2, 5): \[ 5 ; ? < ; 2(2) + \frac{3}{4} \implies 5 < 4 + \frac{3}{4} \implies 5 < 4.75 ; \text{(False)} \]

Conclusion

• Inequality represented by the graph: It is either: \[ y < 2x + \frac{3}{4} \quad (\text{if shaded below}) \quad \text{OR} \quad y > 2x + \frac{3}{4} \quad (\text{if shaded above}) \]

Considering you mentioned "everything to the left of the line is shaded," this reinforces the conclusion that the correct inequality is likely: \[ y < 2x + \frac{3}{4} \]

• Points that satisfy the inequality: All points (-2, -4), (-1, -2), (1, 2) satisfy the inequality \(y < 2x + \frac{3}{4}\).

Final Answer:

• Inequality: \(y < 2x + \frac{3}{4}\)
• Possible solution points: (-2, -4), (-1, -2), (1, 2)

If you need a single point, any of those three can be considered correct as a solution to the inequality.

Answer 8

Which inequality is represented by the graph?

✔ y > 2x + (3/4)



Which point is a solution to the inequality?

✔ (2, 5)

Ms. Cassidy plotted the point (2, 3) on Miguel’s graph ofy < 2x – 4. She instructed him to change one number or one symbol in his inequality so that the point (2, 3) can be included in the solution set.On a coordinate plane, a dashed straight line with positive slope goes through (1, negative 2) and (4, 4). Everything to the right of the line is shaded. Point (2, 3) is also shown. Which equations might Miguel write? Check all that apply.y < 2x – 1y ≤ 2x – 4y > 2x – 4y < 2x + 4y < 3.5x – 4y < 4x – 4

Answer 9

To solve the problem, we first analyze each part step-by-step.

Part 1: Determining the Inequality from the Graph

1. Given Inequality: \[ y > 2x + \frac{3}{4} \] This means that the region above the line is shaded, and the line itself is not included (dashed line).

2. Given Point: The point \((2, 5)\) satisfies this inequality since substituting \(x = 2\) produces: \[ 5 > 2(2) + \frac{3}{4} \rightarrow 5 > 4 + 0.75 \rightarrow 5 > 4.75 \quad \text{(True)} \]

Part 2: Ms. Cassidy's Instructions to Miguel

Ms. Cassidy wants to include the point \((2, 3)\) into Miguel's original inequality \(y < 2x - 4\) by changing "one number or one symbol."

Analyze the Current Inequality

For the original inequality \(y < 2x - 4\):

1. Substitute the point \((2, 3)\): \[ 3 < 2(2) - 4 \rightarrow 3 < 4 - 4 \rightarrow 3 < 0 \quad \text{(False)} \] This shows that \((2, 3)\) is not included in the solution set.

Changes Miguel Might Make

To allow the point \((2, 3)\) to be included, we can either change the inequality symbol from "<" to "≤" or adjust the equation \(2x - 4\) to a larger value at \(x = 2\).

Option 1: Change the Inequality Symbol

1. Changing from \(<\) to \(\leq\):
   • New inequality: \(y \leq 2x - 4\)

Option 2: Change the Equation

2. We could modify the equation:
   • Make the right-hand side larger:
   • For instance, making it larger than \(-(3)\):
   • Any of the forms that satisfy:
     • Change to a constant that allows inclusion:
       • \(y < 2x - 1\) (this will allow the point)
       • Or \(y < 3.5x - 4\) (the slope will limit it enough)

Evaluating Possible Equations

Let's evaluate each option provided to see which might work:

1. Option: \(y < 2x - 1\) -

   • Test point: \((2, 3)\) \[ 3 < 2(2) - 1 \rightarrow 3 < 4 - 1 \rightarrow 3 < 3 \quad \text{(False)} \]

2. Option: \(y \leq 2x - 4\) -

   • Test point: \((2, 3)\) \[ 3 \leq 2(2) - 4 \rightarrow 3 \leq 4 - 4 \rightarrow 3 \leq 0 \quad \text{(False)} \]

3. Option: \(y > 2x - 4\) -

   • Test point: \((2,3)\) \[ 3 > 2(2) - 4 \rightarrow 3 > 4 - 4 \rightarrow 3 > 0 \quad \text{(True)} \]

4. Option: \(y < 2x + 4\) -

   • Test point: \((2, 3)\) \[ 3 < 2(2) + 4 \rightarrow 3 < 4 + 4 \rightarrow 3 < 8 \quad \text{(True)} \]

5. Option: \(y < 3.5x - 4\) -

   • Test point: \((2, 3)\) \[ 3 < 3.5(2) - 4 \rightarrow 3 < 7 - 4 \rightarrow 3 < 3 \quad \text{(False)} \]

6. Option: \(y < 4x - 4\) -

   • Test point: \((2, 3)\) \[ 3 < 4(2) - 4 \rightarrow 3 < 8 - 4 \rightarrow 3 < 4 \quad \text{(True)} \]

Summary of Solutions

Thus, the inequalities Miguel could write to include the point \((2, 3)\) are:

• \(y > 2x - 4\)
• \(y < 2x + 4\)
• \(y < 4x - 4\)

Final Answer:
The inequalities that Miguel might write are:

1. \(y > 2x - 4\)
2. \(y < 2x + 4\)
3. \(y < 4x - 4\)

Only the select options are true based on the evaluations done on the given points.