Use the image to answer the question.

Question

Four points and two intersecting lines are plotted on a coordinate plane. The x and y axes range from negative 5 to 10 in unit increments, labeled at every 5 units.

Which of the points on the graph is a solution for the simultaneous inequalities y>4x−2
and y≥−13x+2
? Enter the x-value followed by the y-value.

Answer 1

To determine which of the points is a solution for the inequalities \( y > 4x - 2 \) and \( y \geq -13x + 2 \), we need to analyze both inequalities:

1. Inequality \( y > 4x - 2 \):

   • This is the equation of a line with a slope of 4 and a y-intercept at \(-2\).
   • The area above this line is the solution region for this inequality.

2. Inequality \( y \geq -13x + 2 \):

   • This is the equation of a line with a slope of -13 and a y-intercept at \(2\).
   • The area above this line (including the line itself) is the solution region for this inequality.

To find the solution set, we identify points that lie in the intersection of the regions defined by both inequalities.

If you have specific points plotted on the coordinate plane, we can evaluate each point to see if it satisfies both inequalities. Here’s how you would check a particular point \((x, y)\):

1. Substitute the x-value into \( y > 4x - 2 \) and check if the y-value is greater than the result.
2. Substitute the x-value into \( y \geq -13x + 2 \) and check if the y-value is greater than or equal to the result.

Please provide the coordinates of the points plotted on the graph, and I can help determine which of them satisfies both inequalities.

Answer 2

the plotted points are (-1, 5), (-4, -2), (5,2) and the point where all the shaded regions connect at the corner is (0.923, 1.6)

Answer 3

Let's evaluate each plotted point to determine whether it satisfies the inequalities \(y > 4x - 2\) and \(y \geq -13x + 2\).

1. Point (-1, 5):

   • For \(y > 4x - 2\): \[ 5 > 4(-1) - 2 \implies 5 > -4 - 2 \implies 5 > -6 \quad \text{(True)} \]
   • For \(y \geq -13x + 2\): \[ 5 \geq -13(-1) + 2 \implies 5 \geq 13 + 2 \implies 5 \geq 15 \quad \text{(False)} \]
   • So, (-1, 5) does not satisfy both inequalities.

2. Point (-4, -2):

   • For \(y > 4x - 2\): \[ -2 > 4(-4) - 2 \implies -2 > -16 - 2 \implies -2 > -18 \quad \text{(True)} \]
   • For \(y \geq -13x + 2\): \[ -2 \geq -13(-4) + 2 \implies -2 \geq 52 + 2 \implies -2 \geq 54 \quad \text{(False)} \]
   • So, (-4, -2) does not satisfy both inequalities.

3. Point (5, 2):

   • For \(y > 4x - 2\): \[ 2 > 4(5) - 2 \implies 2 > 20 - 2 \implies 2 > 18 \quad \text{(False)} \]
   • For \(y \geq -13x + 2\): \[ 2 \geq -13(5) + 2 \implies 2 \geq -65 + 2 \implies 2 \geq -63 \quad \text{(True)} \]
   • Still, (5, 2) does not satisfy both inequalities.

4. Point (0.923, 1.6):

   • For \(y > 4x - 2\): \[ 1.6 > 4(0.923) - 2 \implies 1.6 > 3.692 - 2 \implies 1.6 > 1.692 \quad \text{(False)} \]
   • For \(y \geq -13x + 2\): \[ 1.6 \geq -13(0.923) + 2 \implies 1.6 \geq -11.999 + 2 \implies 1.6 \geq -9.999 \quad \text{(True)} \]
   • So, the point (0.923, 1.6) does not satisfy the first inequality.

Since none of the provided points satisfy both inequalities at the same time, please check if there are points you may have missed or any other relevant information.

From the analysis, no point among the given set satisfies both inequalities \(y > 4x - 2\) and \(y \geq -13x + 2\).

Answer 4

x∈[6−3y,
4
y
​
+
2
1
​
), y>
13
22
​

decipher and help me out

Answer 5

Let's decipher the expression you've provided, which seems to consist of a set notation for \(x\) in terms of \(y\) and a condition on \(y\). Here’s what each part means:

Expression Breakdown:

1. Interval for \(x\): The notation \( x \in [6 - 3y, 4y + 2) \) means that \(x\) can take on values starting from \(6 - 3y\) (included) to \(4y + 2\) (not included).

2. Condition on \(y\): The condition \(y > \frac{13}{22}\) means \(y\) must be greater than approximately \(0.5909\).

What This Means:

• The values of \(x\) depend on the value of \(y\), which must be greater than \( \frac{13}{22} \).
• Depending on the value of \(y\), you can calculate the lower and upper bounds for \(x\).

Example Calculation:

1. Determine \(x\) bounds for a specific \(y\):

   • If you choose, for example, \(y = 1\):
     • Lower bound: \(6 - 3(1) = 6 - 3 = 3\)
     • Upper bound: \(4(1) + 2 = 4 + 2 = 6\)
   • Thus, for \(y = 1\), the interval for \(x\) is \(x \in [3, 6)\).

2. You can repeat this with any \(y > \frac{13}{22}\). For instance:

   • If \(y = 2\):
     • Lower bound: \(6 - 3(2) = 6 - 6 = 0\)
     • Upper bound: \(4(2) + 2 = 8 + 2 = 10\)
   • For \(y = 2\), the interval for \(x\) is \(x \in [0, 10)\).

Summary:

You can choose any \(y\) value greater than \( \frac{13}{22} \) and calculate the corresponding \(x\) range as \(x \in [6 - 3y, 4y + 2)\). If you need help solving for specific values or a particular context, please provide that additional information!

Answer 6

and here is y
{
y≥−
3
x
​
+2,
y>4x−2,
​

x<
13
12
​

x≥
13
12

Answer 7

Let's break down the information you've provided clearly and see how it integrates into solving for \(y\) while observing the conditions on \(x\). It looks like you have a set of inequalities involving both \(x\) and \(y\).

Inequalities Breakdown

1. Inequalities for \(y\):

   • \(y \geq -3x + 2\)
   • \(y > 4x - 2\)

2. Conditions for \(x\):

   • \(x < \frac{13}{12}\)
   • \(x \geq \frac{13}{12}\)

Analyzing the Conditions

1. The inequalities for \(y\) represent two lines:

   • Line 1: \(y = -3x + 2\) (a line with a slope of \(-3\) and y-intercept \(2\))
   • Line 2: \(y = 4x - 2\) (a line with a slope of \(4\) and y-intercept \(-2\))

   The area above the first line (included) corresponds to \(y \geq -3x + 2\), and the area above the second line (not included) corresponds to \(y > 4x - 2\).

2. The conditions for \(x\) are a little contradictory. You have:

   • \(x < \frac{13}{12}\) and \(x \geq \frac{13}{12}\)

   However, these two conditions cannot be true simultaneously. One is a strict inequality (less than) and the other is inclusive (greater than or equal to).

Evaluating for \(x\)

Since the conditions for \(x\) are contradictory, you will either need to clarify the conditions or work with one of these cases.

Case 1: If we only consider \(x < \frac{13}{12}\)

If we only consider \(x < \frac{13}{12}\), you can plug values of \(x\) that fall under this condition into the inequalities for \(y\) to find corresponding values.

For example:

• Let’s choose \(x = 1\) (which is less than \(\frac{13}{12}\)):

  • \(y \geq -3(1) + 2 = -3 + 2 = -1\) (so \(y\) must be greater than or equal to \(-1\))
  • \(y > 4(1) - 2 = 4 - 2 = 2\) (so \(y\) must be greater than \(2\))

  Here, \(y\) must satisfy both conditions. Since \(y > 2\) is stricter than \(y \geq -1\), we conclude:

  • For \(x = 1\): \(y > 2\)

Case 2: If we consider \(x \geq \frac{13}{12}\)

Since having both inequalities for \(x\ doesn't work simultaneously, use just (x < \frac{13}{12}\) or \(x \geq \frac{13}{12}\) to explore possibilities.

• Since \(x \geq \frac{13}{12}\) would generally lead to a single point or boundary,
• If we take \(x = \frac{13}{12}\):
  • \(y \geq -3\left(\frac{13}{12}\right) + 2 = -\frac{39}{12} + 2 = -\frac{39}{12} + \frac{24}{12} = -\frac{15}{12}\)
  • \(y > 4\left(\frac{13}{12}\right) - 2 = \frac{52}{12} - 2 = \frac{52}{12} - \frac{24}{12} = \frac{28}{12} = \frac{7}{3}\)

Summary

• If considering \(x < \frac{13}{12}\): You can find \((x,y)\) pairs that satisfy both conditions on \(y\).
• If considering \(x \geq \frac{13}{12}\): There is a single point pair where \(x = \frac{13}{12}\) and \(y \geq -\frac{15}{12}\), and \(y > \frac{7}{3}\).

Please clarify how you’d like to proceed with the contradictions in the conditions for \(x\) or if you'd like to focus on a specific range!

Answer 8

the shaded regions intersect at 0.923, 1.6

Answer 9

Given the point \((0.923, 1.6)\) where the shaded regions (defined by the inequalities you provided) intersect, we can verify whether this point satisfies the inequalities for \(y\).

Re-evaluation of Inequalities

You specified:

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

Checking the Point \((0.923, 1.6)\)

1. Check the first inequality: \(y \geq -3x + 2\)

   • Substitute \(x = 0.923\) and \(y = 1.6\): \[ 1.6 \geq -3(0.923) + 2 \]
   • Calculate the right side: \[ -3(0.923) = -2.769 \quad \text{and thus} \quad -2.769 + 2 = -0.769 \]
   • Therefore, we check: \[ 1.6 \geq -0.769 \quad \text{(True)} \]

2. Check the second inequality: \(y > 4x - 2\)

   • Substitute \(x = 0.923\) and \(y = 1.6\): \[ 1.6 > 4(0.923) - 2 \]
   • Calculate the right side: \[ 4(0.923) = 3.692 \quad \text{and thus} \quad 3.692 - 2 = 1.692 \]
   • So, we check: \[ 1.6 > 1.692 \quad \text{(False)} \]

Conclusion

The point \((0.923, 1.6)\) satisfies the first inequality \(y \geq -3x + 2\) but does not satisfy the second inequality \(y > 4x - 2\).

If we review the intersection areas corresponding to the inequalities:

• Where \(y = -3x + 2\) (lower boundary) gives a feasible area above the line.
• Where \(y = 4x - 2\) (upper boundary) gives a feasible area below the line.

Finding the Actual Intersection Points

You can also find the intersection of the two lines defined by the inequalities to locate other critical points if required:

Set the two equations equal to each other: \[ -3x + 2 = 4x - 2 \]

Solving for \(x\): \[ 2 + 2 = 4x + 3x \ 4 = 7x \ x = \frac{4}{7} \approx 0.5714 \]

Now find \(y\) by substituting \(x\) back into either equation: \[ y = -3\left(\frac{4}{7}\right) + 2 \ y = -\frac{12}{7} + \frac{14}{7} = \frac{2}{7} \approx 0.2857 \]

Thus the intersection point of the lines is \(\left(\frac{4}{7}, \frac{2}{7}\right)\).

Summary

• The point \((0.923, 1.6)\) does not satisfy both inequalities.
• The actual intersection between the two lines (where both conditions hold) is \(\left(\frac{4}{7}, \frac{2}{7}\right) \approx (0.5714, 0.2857)\).

Always ensure the intersections and points reflect the conditions outlined by the inequalities when analyzing shaded regions or solution spaces. If you have more parts of the problem or other points to verify, feel free to share!