Which is the graph of the solution set of −2x + 5y 15? 1. On a coordinate plane, a dashed straight line has a positive slope and goes through (0, 3) and (10, 7). Everything above and to the left of the line is shaded. 2. On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 3) and (5, 1). Everything below and to the left of the line is shaded. 3. On a coordinate plane, a dashed straight line has a positive slope and goes through (0, 5) and (10, 9). Everything above and to the left of the line is shaded.

Question

Which is the graph of the solution set of −2x + 5y > 15? 1. On a coordinate plane, a dashed straight line has a positive slope and goes through (0, 3) and (10, 7). Everything above and to the left of the line is shaded. 2. On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 3) and (5, 1). Everything below and to the left of the line is shaded. 3. On a coordinate plane, a dashed straight line has a positive slope and goes through (0, 5) and (10, 9). Everything above and to the left of the line is shaded.

GPT-4o mini · Accepted Answer

To analyze the inequality $-2x + 5y > 15$, we first convert the inequality into an equation to find the corresponding line: $$ -2x + 5y = 15 $$ Rearranging this gives us: $$ 5y = 2x + 15 $$ $$ y = \frac{2}{5}x + 3 $$ This line has a positive slope of $\frac{2}{5}$ and a y-intercept at $3$. Now, we need to determine the region that satisfies the inequality $ -2x + 5y > 15$. Since this is a "greater than" inequality, we will shade the region above the line (as $y > \frac{2}{5}x + 3$). Next, let's evaluate the options: 1. **First option:** A dashed line with a positive slope and going through $(0, 3)$ and $(10, 7)$. This fits our line's description, and the shaded region above and to the left indicates $y > \frac{2}{5}x + 3$, which matches our requirement. 2. **Second option:** A dashed line with a negative slope going through $(0, 3)$ and $(5, 1)$. This does not have a positive slope and does not match our line. 3. **Third option:** A dashed line with a positive slope going through $(0, 5)$ and $(10, 9)$. This line's intercept (5) is higher than ours (3); therefore, it does not represent our solution set as it is possibly for a different inequality. Based on this analysis, the only graph that correctly represents the solution set of the inequality $-2x + 5y > 15$ is: **Option 1.**

Answer

On a coordinate plane, two lines are shown. The first dashed line has a positive slope and goes through (negative 8, 0) and (0, 3). Everything below and to the right of the line is shaded. The second solid line has a negative slope and goes through (negative 5, 0) and (0, negative 2). Everything above and to the right of the line is shaded. Which ordered pairs make both inequalities true? Check all that apply.(−5, 5)
(0, 3)
(0, −2)
(1, 1)
(3, −4)

GPT-4o mini · Answer

To determine which ordered pairs satisfy both inequalities defined by the two lines, we first need the equations for each line based on the points given.

First Line (Dashed):

The first line goes through the points $(-8, 0)$ and $(0, 3)$.

Finding the slope (m): \[ m = \frac{3 - 0}{0 - (-8)} = \frac{3}{8} \]
Finding the equation of the line: Using the point-slope form $y - y_1 = m(x - x_1)$: \[ y - 0 = \frac{3}{8}(x - (-8)) \] \[ y = \frac{3}{8}x + 3 \]

Since this line is dashed and the region below and to the right is shaded, the inequality is: \[ y < \frac{3}{8}x + 3 \]

Second Line (Solid):

The second line goes through the points $(-5, 0)$ and $(0, -2)$.

Finding the slope (m): \[ m = \frac{-2 - 0}{0 - (-5)} = \frac{-2}{5} \]
Finding the equation of the line: Using the point-slope form: \[ y - 0 = \frac{-2}{5}(x - (-5)) \] \[ y = \frac{-2}{5}x - 2 \]

Since this line is solid and the region above and to the right is shaded, the inequality is: \[ y \geq \frac{-2}{5}x - 2 \]

Testing the Ordered Pairs:

Now we'll check each ordered pair against both inequalities.

Pair: $(-5, 5)$
- For the first inequality: \[ 5 < \frac{3}{8}(-5) + 3 \implies 5 < -\frac{15}{8} + 3 \implies 5 < \frac{24 - 15}{8} \implies 5 < \frac{9}{8} \text{ (False)} \]
- This pair does not satisfy the first inequality.
Pair: $(0, 3)$
- For the first inequality: \[ 3 < 3 \text{ (False)} \]
- This pair does not satisfy the first inequality.
Pair: $(0, -2)$
- For the first inequality: \[ -2 < 3 \text{ (True)} \]
- For the second inequality: \[ -2 \geq -2 \text{ (True)} \]
- This pair satisfies both inequalities.
Pair: $(1, 1)$
- For the first inequality: \[ 1 < \frac{3}{8}(1) + 3 \implies 1 < \frac{3}{8} + 3 \implies 1 < \frac{3 + 24}{8} \implies 1 < \frac{27}{8} \text{ (True)} \]
- For the second inequality: \[ 1 \geq \frac{-2}{5}(1) - 2 \implies 1 \geq -\frac{2}{5} - 2 \implies 1 \geq -\frac{2 + 10}{5} \implies 1 \geq -\frac{12}{5} \text{ (True)} \]
- This pair satisfies both inequalities.
Pair: $(3, -4)$
- For the first inequality: \[ -4 < \frac{3}{8}(3) + 3 \implies -4 < \frac{9}{8} + 3 \implies -4 < \frac{9 + 24}{8} \implies -4 < \frac{33}{8} \text{ (True)} \]
- For the second inequality: \[ -4 \geq \frac{-2}{5}(3) - 2 \implies -4 \geq -\frac{6}{5} - 2 \implies -4 \geq -\frac{6 + 10}{5} \implies -4 \geq -\frac{16}{5} \text{ (False)} \]
- This pair does not satisfy the second inequality.

Summary:

The pairs that satisfy both inequalities are:

(0, -2)
(1, 1)

So, the ordered pairs that make both inequalities true are:

(0, -2)
(1, 1)

Answer

On a coordinate plane, a solid straight line has a negative slope and goes through (negative 2, 2) and (2, negative 8). Everything above and to the right of the line is shaded. The line has an equation of y greater-than-or-equal-to negative five-halves x minus 3.Which linear equality will not have a shared solution set with the graphed linear inequality?y > Two-fifthsx + 2y < Negative five-halvesx – 7y > Negative two-fifthsx – 5y < Five-halvesx + 2

GPT-4o mini · Answer

To determine which of the given linear inequalities will not have a shared solution set with the given graph, we should analyze the boundary line and the shaded area of the provided inequality.

The given equation is:

\[ y \geq -\frac{5}{2}x - 3 \]

Step 1: Analyze the Given Inequality

The line $y = -\frac{5}{2}x - 3$ has:
- A negative slope of $-\frac{5}{2}$
- A y-intercept of $-3$

Since the inequality is greater than or equal to, the region shaded is above this line, indicating that the solutions to the inequality are in the area of the Cartesian plane where $y$ values are above the line.

Step 2: Analyze Each Option

Now let's analyze each of the proposed inequalities to see where they fall relative to the boundary line.

Option 1: $y > \frac{2}{5}x + 2$

This inequality represents a line with a positive slope of $\frac{2}{5}$ and a y-intercept of $2$. The region for this inequality is above this line. This line will intersect the line given by $y \geq -\frac{5}{2}x - 3$ in some areas of the graph, so it may have a shared solution set.
Option 2: $y < -\frac{5}{2}x - 7$

This line also has a negative slope of $-\frac{5}{2}$ but has a y-intercept of $-7$. The shading will be below this line. Since this line is below (and matches in slope) the original line $y \geq -\frac{5}{2}x - 3$ but at a lower point (y-intercept), these two inequalities cannot share a solution set.
Option 3: $y > -\frac{2}{5}x - 5$

This line also has a negative slope of $-\frac{2}{5}$ and a y-intercept of $-5$. Similar to option 1, this will also create a region of solutions above its line, which may overlap with solutions from the initial inequality.
Option 4: $y < \frac{5}{2}x + 2$

This inequality represents a line with a positive slope of $\frac{5}{2}$ and a y-intercept of $2$. The shading will be below this line. There may be some overlap above the line $y \geq -\frac{5}{2}x - 3$, making it potentially have a shared solution.

Conclusion:

The inequality that will not have a shared solution set with the graphed linear inequality is:

\[ y < -\frac{5}{2}x - 7 \]

This is because this inequality represents a region entirely below the boundary line of the provided inequality, while the provided inequality solutions are specifically above it.

Answer

Given the system of inequalities: 4x – 5y < 1One-halfy – x < 3 Which shows the given inequalities in slope-intercept form?y < Four-fifthsx – One-fifthy < 2x + 6y > Four-fifthsx – One-fifthsy < 2x + 6y > Negative four-fifthsx + One-fifthy > 2x + 6

GPT-4o mini · Answer

To convert the given inequalities to slope-intercept form (y = mx + b), we'll manipulate each inequality step by step.

Step 1: Convert the first inequality to slope-intercept form

Inequality 1:
\[ 4x - 5y < 1 \]

Subtract $4x$ from both sides: \[-5y < -4x + 1\]
Divide each term by $-5$, and remember to reverse the inequality sign: \[y > \frac{4}{5}x - \frac{1}{5}\]

Step 2: Convert the second inequality to slope-intercept form

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

Add $x$ to both sides: \[\frac{1}{2}y < x + 3\]
Multiply each term by $2$ to eliminate the fraction: \[y < 2x + 6\]

Summary of the inequalities in slope-intercept form:

First inequality: \[y > \frac{4}{5}x - \frac{1}{5}\]
Second inequality: \[y < 2x + 6\]

Analyzing the options listed:

$y < \frac{4}{5}x - \frac{1}{5}$ (Incorrect, the first inequality is $y > \frac{4}{5}x - \frac{1}{5}$)
$y < 2x + 6$ (Correct for the second inequality)
$y > \frac{4}{5}x - \frac{1}{5}$ (Correct for the first inequality)
$y < 2x + 6$ (Same as previously mentioned for the second)
$y > -\frac{4}{5}x + \frac{1}{5}$ (Incorrect; this is the opposite and not part of the original inequalities)
$y > 2x + 6$ (Incorrect)

The correct matching inequalities in slope-intercept form from the analysis:

First inequality: $y > \frac{4}{5}x - \frac{1}{5}$
Second inequality: $y < 2x + 6$

Hence:

From the options provided, the correct representation showing the inequalities in slope-intercept form are:

$y > \frac{4}{5}x - \frac{1}{5}$
$y < 2x + 6$

If this specific format is not listed among your options, please confirm the options for exact phrasing; otherwise, the inequalities converted as detailed above would be correct!

Answer

Which is the graph of the system of inequalities y > Four-fifthsx – One-fifth and y < 2x + 6 1. 1. On a coordinate plane, 2 lines are shown. The first solid straight line has a positive slope and goes through (negative 3, 0) and (0, 6). Everything to the right of the line is shaded. The second solid straight line has a positive slope and goes through (negative 2, negative 2) and (4, 3). Everything above and to the left of the line is shaded. 2. On a coordinate plane, 2 lines are shown. The first solid straight line has a negative slope and goes through (negative 4, 3) and (2, negative 2). Everything to the right of the line is shaded. The second solid straight line has a negative slope and goes through (0, 6) and (3, 0). Everything to the left of the line is shaded. 3. On a coordinate plane, 2 lines are shown. The first solid straight line has a positive slope and goes through (negative 4, negative 3) and (2, 2). Everything above and to the left of the line is shaded. The second solid straight line has a positive slope and goes through (0, negative 6) and (3, 0). Everything to the right of the line is shaded. 4. On a coordinate plane, 2 lines are shown. The first solid straight line has a positive slope and goes through (negative 3, 0) and (0, 6). Everything to the left of the line is shaded. The second solid straight line has a positive slope and goes through (negative 2, negative 2) and (4, 3). Everything below and to the right of the line is shaded.

GPT-4o mini · Answer

To find the correct graph of the system of inequalities $y > \frac{4}{5}x - \frac{1}{5}$ and $y < 2x + 6$, we'll analyze the lines and the respective shaded regions.

Step 1: Analyze the Inequalities

First Inequality: $y > \frac{4}{5}x - \frac{1}{5}$
- This line has a positive slope of $\frac{4}{5}$ and a y-intercept of $-\frac{1}{5}$ (which is very close to 0).
- The region that satisfies this inequality is above the line since it is a "greater than" inequality.
Second Inequality: $y < 2x + 6$
- The line has a positive slope of $2$ and a y-intercept of $6$.
- The region that satisfies this inequality is below the line since it is a "less than" inequality.

Step 2: Intersecting Regions

The solution to the system of inequalities will be the area that is above the first line and below the second line.

Step 3: Analyze the Given Graphs

Option 1:
- First line is solid and has a positive slope, shading to the right. This suggests $y \geq \text{line}$ instead of $y > \text{line}$.
- Second line is also solid with positive slope, shading above and to the left. This does not match $y < \text{line}$.
Option 2:
- Both lines have negative slopes and do not match the given inequalities which have positive slopes. Therefore, this option is not correct.
Option 3:
- The first line has a positive slope with shading above and to the left (does not match $y > \frac{4}{5}x - \frac{1}{5}$).
- The second line has a positive slope with shading to the right (incorrect).
Option 4:
- First line has a positive slope with shading to the left (indicating $y \leq \text{line}$, incorrect for $y > \frac{4}{5}x - \frac{1}{5}$).
- The second line has a positive slope with shading below and to the right.

Conclusion

Given the analysis of the inequalities and the options provided:

None of the options presented matches the correct representation of the system $y > \frac{4}{5}x - \frac{1}{5}$ (above the line) and $y < 2x + 6$ (below the line).

If there’s a miscommunication or mislabeling in the description of the options, kindly double-check them or share more specific information about the lines and shading for a more precise conclusion. Based on the current descriptions, none of the provided graphs align with the corresponding regions of the given inequalities.

Answer

option 1. On a coordinate plane, 2 lines are shown. The first solid straight line has a positive slope and goes through (negative 3, 0) and (0, 6). Everything to the right of the line is shaded. The second solid straight line has a positive slope and goes through (negative 2, negative 2) and (4, 3). Everything above and to the left of the line is shaded.

the graph that has two lines going up at the up right corner and the lines connecting at the bottome left shaded on top orange grey and blue.

Solve the system of inequalities: y + 2x > 3 and y Greater-than-or-equal-to 3.5x − 5

The first inequality, y + 2x > 3, is___
in slope-intercept form.

The first inequality, y + 2x > 3, has a___
boundary line.

The second inequality, y Greater-than-or-equal-to 3.5x − 5, has a___
boundary line.

Both inequalities have a solution set that is shaded___
their boundary lines.

___is a point in the solution set of the system of inequalities.

GPT-4o mini · Answer

Let's solve the given system of inequalities step by step.

Step 1: Convert the First Inequality to Slope-Intercept Form

First Inequality: \[ y + 2x > 3 \]

Subtract $2x$ from both sides: \[ y > -2x + 3 \]

So the first inequality in slope-intercept form is: \[ y > -2x + 3 \]

Step 2: Analyze the First Inequality

The first inequality has:

A boundary line that is dashed (since it is a "greater than" inequality).

Step 3: Write the Second Inequality in Slope-Intercept Form

Second Inequality: \[ y \geq 3.5x - 5 \]

This inequality is already in slope-intercept form. Here, the slope is $3.5$ and the y-intercept is $-5$.

Step 4: Analyze the Second Inequality

The second inequality has:

A boundary line that is solid (since it is a "greater than or equal to" inequality).

Step 5: Analyze the Shading for Both Inequalities

For the first inequality $y > -2x + 3$:
- The solution set is shaded above the line.
For the second inequality $y \geq 3.5x - 5$:
- The solution set is shaded above the line as well.

Summary of Findings

The first inequality, $y + 2x > 3$, is in slope-intercept form: \[ y > -2x + 3 \]
The first inequality has a dashed boundary line.
The second inequality has a solid boundary line.
Both inequalities have a solution set that is shaded above their boundary lines.

Step 6: Identify a Point in the Solution Set

To find a point in the solution set, we can check points above both lines. Let's choose a simple point.

A reasonable point to check would be $ (0, 4) $:

Check $y + 2x > 3$: \[ 4 + 2(0) > 3 \implies 4 > 3 \] (True)
Check $y \geq 3.5x - 5$: \[ 4 \geq 3.5(0) - 5 \implies 4 \geq -5 \] (True)

Since $ (0, 4) $ satisfies both inequalities, it is a point in the solution set of the system of inequalities.

Final Answer

The first inequality, $ y + 2x > 3 $, is in slope-intercept form: $ y > -2x + 3 $.
The first inequality has a dashed boundary line.
The second inequality has a solid boundary line.
Both inequalities have a solution set that is shaded above their boundary lines.
$ (0, 4) $ is a point in the solution set of the system of inequalities.