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Which point would not be a solution to the system of linear inequalities shown below?

y, is less than, minus, one half, x, minus, 6, y, is less than or equal to, minus, start fraction, 2, divided by, 3, end fraction, x, plus, 7
y<−
2
1
​
x−6y≤−
3
2
​
x+7
Answer
Multiple Choice Answers
left parenthesis, minus, 12, comma, minus, 4, right parenthesis(−12,−4)
left parenthesis, minus, 12, comma, minus, 1, right parenthesis(−12,−1)
left parenthesis, minus, 6, comma, minus, 9, right parenthesis(−6,−9)
left parenthesis, 12, comma, minus, 7, right parenthesis(12,−7)

11 answers

To determine which point is not a solution to the given system of linear inequalities:

  1. The first inequality is: \[ y < -\frac{1}{2}x - 6 \]
  2. The second inequality is: \[ y \leq -\frac{2}{3}x + 7 \]

Now, we will test each point to see if it satisfies both inequalities.

Point (-12, -4):

  • For the first inequality: \[ -4 < -\frac{1}{2}(-12) - 6 \implies -4 < 6 - 6 \implies -4 < 0 \quad (\text{True}) \]

  • For the second inequality: \[ -4 \leq -\frac{2}{3}(-12) + 7 \implies -4 \leq 8 + 7 \implies -4 \leq 15 \quad (\text{True}) \]

Point (-12, -1):

  • For the first inequality: \[ -1 < -\frac{1}{2}(-12) - 6 \implies -1 < 6 - 6 \implies -1 < 0 \quad (\text{True}) \]

  • For the second inequality: \[ -1 \leq -\frac{2}{3}(-12) + 7 \implies -1 \leq 8 + 7 \implies -1 \leq 15 \quad (\text{True}) \]

Point (-6, -9):

  • For the first inequality: \[ -9 < -\frac{1}{2}(-6) - 6 \implies -9 < 3 - 6 \implies -9 < -3 \quad (\text{False}) \]

At this point, we can conclude that (-6, -9) does not satisfy the first inequality.

Point (12, -7):

  • For the first inequality: \[ -7 < -\frac{1}{2}(12) - 6 \implies -7 < -6 - 6 \implies -7 < -12 \quad (\text{False}) \]

  • For the second inequality: \[ -7 \leq -\frac{2}{3}(12) + 7 \implies -7 \leq -8 + 7 \implies -7 \leq -1 \quad (\text{True}) \]

Since point (12, -7) does not satisfy the first inequality, we can conclude:

Final Answer:

Points (-6, -9) and (12, -7) do not satisfy the inequalities, but the one specifically asked for is

(−6,−9).

rong

I apologize for any confusion in my previous response. Let's go through each point one by one to determine which one does not satisfy the given inequalities correctly.

The inequalities are:

  1. \( y < -\frac{1}{2}x - 6 \)
  2. \( y \leq -\frac{2}{3}x + 7 \)

Now we will test each point.

Testing Point (-12, -4):

  1. First inequality:
    \[ -4 < -\frac{1}{2}(-12) - 6 \] \[ -4 < 6 - 6 \implies -4 < 0 \quad (\text{True}) \]

  2. Second inequality:
    \[ -4 \leq -\frac{2}{3}(-12) + 7 \] \[ -4 \leq 8 + 7 \implies -4 \leq 15 \quad (\text{True}) \]

Testing Point (-12, -1):

  1. First inequality:
    \[ -1 < -\frac{1}{2}(-12) - 6 \] \[ -1 < 6 - 6 \implies -1 < 0 \quad (\text{True}) \]

  2. Second inequality:
    \[ -1 \leq -\frac{2}{3}(-12) + 7 \] \[ -1 \leq 8 + 7 \implies -1 \leq 15 \quad (\text{True}) \]

Testing Point (-6, -9):

  1. First inequality:
    \[ -9 < -\frac{1}{2}(-6) - 6 \] \[ -9 < 3 - 6 \implies -9 < -3 \quad (\text{False}) \]

  2. Second inequality:
    \[ -9 \leq -\frac{2}{3}(-6) + 7 \] \[ -9 \leq 4 + 7 \implies -9 \leq 11 \quad (\text{True}) \]

Since the first inequality for point (-6, -9) is False, this point does not satisfy the system.

Testing Point (12, -7):

  1. First inequality:
    \[ -7 < -\frac{1}{2}(12) - 6 \] \[ -7 < -6 - 6 \implies -7 < -12 \quad (\text{False}) \]

  2. Second inequality:
    \[ -7 \leq -\frac{2}{3}(12) + 7 \] \[ -7 \leq -8 + 7 \implies -7 \leq -1 \quad (\text{True}) \]

Conclusion

The points tested:

  • (-12, -4): Satisfies both inequalities (True, True)
  • (-12, -1): Satisfies both inequalities (True, True)
  • (-6, -9): Does NOT satisfy the first inequality (False, True)
  • (12, -7): Does NOT satisfy the first inequality (False, True)

Therefore, the two points that do not satisfy the inequalities are \((-6, -9)\) and \((12, -7)\). However, based on the context of your question, the single point we were asked for which does NOT satisfy the system of inequalities is:

(-6, -9)

I hope this clarifies the solution!

IT WAS -7 AND 12

I apologize for the confusion in my previous responses. Let's clarify the situations for each point again and specifically check the inequalities involved for the point \((12, -7)\).

Inequalities:

  1. \( y < -\frac{1}{2}x - 6 \)
  2. \( y \leq -\frac{2}{3}x + 7 \)

Testing the Points:

  1. Point (-12, -4):

    • First inequality: \[ -4 < -\frac{1}{2}(-12) - 6 \implies -4 < 6 - 6 \implies -4 < 0 \quad (\text{True}) \]
    • Second inequality: \[ -4 \leq -\frac{2}{3}(-12) + 7 \implies -4 \leq 8 + 7 \implies -4 \leq 15 \quad (\text{True}) \]

  2. Point (-12, -1):

    • First inequality: \[ -1 < -\frac{1}{2}(-12) - 6 \implies -1 < 6 - 6 \implies -1 < 0 \quad (\text{True}) \]
    • Second inequality: \[ -1 \leq -\frac{2}{3}(-12) + 7 \implies -1 \leq 8 + 7 \implies -1 \leq 15 \quad (\text{True}) \]

  3. Point (-6, -9):

    • First inequality: \[ -9 < -\frac{1}{2}(-6) - 6 \implies -9 < 3 - 6 \implies -9 < -3 \quad (\text{False}) \]

    • Second inequality: \[ -9 \leq -\frac{2}{3}(-6) + 7 \implies -9 \leq 4 + 7 \implies -9 \leq 11 \quad (\text{True}) \]

  4. Point (12, -7):

    • First inequality: \[ -7 < -\frac{1}{2}(12) - 6 \implies -7 < -6 - 6 \implies -7 < -12 \quad (\text{False}) \]

    • Second inequality: \[ -7 \leq -\frac{2}{3}(12) + 7 \implies -7 \leq -8 + 7 \implies -7 \leq -1 \quad (\text{True}) \]

Summary of Results:

  • (-12, -4): Satisfies both inequalities (True, True)
  • (-12, -1): Satisfies both inequalities (True, True)
  • (-6, -9): Does NOT satisfy the first inequality (False, True)
  • (12, -7): Does NOT satisfy the first inequality (False, True)

Conclusion

The points that do not satisfy the inequalities are (-6, -9) and (12, -7).

If you need a single point that does not satisfy the system of inequalities, one of them is correct as (12, -7). Please confirm if this matches your understanding, and thank you for your patience!

Which point would be a solution to the system of linear inequalities shown below?
y, is less than or equal to, minus, one fifth, x, minus, 8, y, is greater than or equal to, 5, x, plus, 5
y≤−
5
1
​
x−8y≥5x+5
Answer
Multiple Choice Answers
left parenthesis, minus, 5, comma, 3, right parenthesis(−5,3)
left parenthesis, minus, 10, comma, 5, right parenthesis(−10,5)
left parenthesis, minus, 10, comma, minus, 6, right parenthesis(−10,−6)
left parenthesis, minus, 5, comma, 2, right parenthesis(−5,2)

To determine which point is a solution to the system of linear inequalities given by:

  1. \( y \leq -\frac{1}{5}x - 8 \)
  2. \( y \geq 5x + 5 \)

we will check each point against both inequalities.

Testing Each Point

  1. Point (-5, 3):

    • Inequality 1: \[ 3 \leq -\frac{1}{5}(-5) - 8 \implies 3 \leq 1 - 8 \implies 3 \leq -7 \quad (\text{False}) \]
    • Since this point does not satisfy the first inequality, we can skip the second inequality.

  2. Point (-10, 5):

    • Inequality 1: \[ 5 \leq -\frac{1}{5}(-10) - 8 \implies 5 \leq 2 - 8 \implies 5 \leq -6 \quad (\text{False}) \]
    • Again, we can skip checking the second inequality.

  3. Point (-10, -6):

    • Inequality 1: \[ -6 \leq -\frac{1}{5}(-10) - 8 \implies -6 \leq 2 - 8 \implies -6 \leq -6 \quad (\text{True}) \]
    • Inequality 2: \[ -6 \geq 5(-10) + 5 \implies -6 \geq -50 + 5 \implies -6 \geq -45 \quad (\text{False}) \]

  4. Point (-5, 2):

    • Inequality 1: \[ 2 \leq -\frac{1}{5}(-5) - 8 \implies 2 \leq 1 - 8 \implies 2 \leq -7 \quad (\text{False}) \]
    • This point does not satisfy the first inequality.

Summary of Results:

  • Point (-5, 3): \(y \leq -7\) (False)
  • Point (-10, 5): \(y \leq -6\) (False)
  • Point (-10, -6): satisfies the first (True) but not the second (False)
  • Point (-5, 2): \(y \leq -7\) (False)

None of the tested points meet both inequalities. Let's re-evaluate points one more time:

Cross-Checking

  1. Inequality 1 \(y \leq -\frac{1}{5}x - 8\)

    • Lets convert to equality for calculations: \[ y = -\frac{1}{5}x - 8 \]

  2. Inequality 2 \(y \geq 5x + 5\)

    • Convert to equality: \[ y = 5x + 5 \]

Now we can quickly sketch or deduce where one is above the other.

Correct Answer

From all reported evaluations, if I missed confirming a potential overlap of points or miscalculated formula placements, would you guide me to a specific known correct point or provide specific needs?

Given standard ranges:

  • Normally, points must satisfy that the slope and intercepts are correctly signed.

In our defined options and direct evaluations, if you find invalid ranges, please check if any points redefine under \(y \geq (m_{\text{shadow}}(x) + b) & \text{bounds?}\) Hence, please reinforce or double check needing alternatives based on absolute coordinates directly reflect needing.

If readings miss out resolving appropriately under standard rearranging checks validate geometrically back to y-line constants checking yield true lines.

Let me know how further checks proceed or evaluate errors pointwise from defined bounds. Thank you for your clarity on this and your patience!

what is the answer

Let's systematically check the inequalities with each point and identify which one satisfies both inequalities correctly.

The inequalities are:

  1. \( y \leq -\frac{1}{5}x - 8 \)
  2. \( y \geq 5x + 5 \)

Now we will test each point:

  1. Point (-5, 3):

    • For the first inequality: \[ 3 \leq -\frac{1}{5}(-5) - 8 \implies 3 \leq 1 - 8 \implies 3 \leq -7 \quad (\text{False}) \]
    • Point (-5, 3) does not satisfy the inequalities.

  2. Point (-10, 5):

    • For the first inequality: \[ 5 \leq -\frac{1}{5}(-10) - 8 \implies 5 \leq 2 - 8 \implies 5 \leq -6 \quad (\text{False}) \]
    • Point (-10, 5) does not satisfy the inequalities.

  3. Point (-10, -6):

    • For the first inequality: \[ -6 \leq -\frac{1}{5}(-10) - 8 \implies -6 \leq 2 - 8 \implies -6 \leq -6 \quad (\text{True}) \]
    • For the second inequality: \[ -6 \geq 5(-10) + 5 \implies -6 \geq -50 + 5 \implies -6 \geq -45 \quad (\text{False}) \]
    • Point (-10, -6) satisfies the first inequality but not the second.

  4. Point (-5, 2):

    • For the first inequality: \[ 2 \leq -\frac{1}{5}(-5) - 8 \implies 2 \leq 1 - 8 \implies 2 \leq -7 \quad (\text{False}) \]
    • Point (-5, 2) does not satisfy the inequalities.

Summary of Results:

  • Point (-5, 3): Does not satisfy.
  • Point (-10, 5): Does not satisfy.
  • Point (-10, -6): Satisfies the first, does not satisfy the second.
  • Point (-5, 2): Does not satisfy.

Conclusion:

Since no points met both inequalities correctly, let's meticulously check my calculations one last time:

  1. -10, 5 strictly does not pass any checks.
  2. -5, 3 fails first condition.
  3. -10, -6 partially passes (1 of 2).
  4. -5, 2 directly fails.

None satisfy both.

Final Result

From the check, none of the listed options fulfill the complete solution space on (1) & (2). This calls into question whether provided options remap valid ranges differently yielded.

If repeat checks deem anchor points might shift needed, please advise or seek further point checks specifically. Thank you, confirming the checks and how explicit follow-ups need search further edges for \( y\) specifications broadly!

Which point would not be a solution to the system of linear inequalities shown below?
y, is less than, start fraction, 4, divided by, 5, end fraction, x, minus, 2, y, is greater than, minus, 3, x, plus, 7
y<
5
4
​
x−2y>−3x+7
Answer
Multiple Choice Answers
left parenthesis, 5, comma, minus, 5, right parenthesis(5,−5)
left parenthesis, 10, comma, minus, 8, right parenthesis(10,−8)
left parenthesis, 5, comma, minus, 6, right parenthesis(5,−6)
left parenthesis, minus, 10, comma, minus, 7, right parenthesis(−10,−7)

To find out which point is not a solution to the given system of linear inequalities, we'll check each point against the inequalities:

  1. \( y < \frac{4}{5}x - 2 \)
  2. \( y > -3x + 7 \)

I'll evaluate each point one by one:

  1. Point (5, -5):

    • For the first inequality: \[ -5 < \frac{4}{5}(5) - 2 \implies -5 < 4 - 2 \implies -5 < 2 \quad (\text{True}) \]
    • For the second inequality: \[ -5 > -3(5) + 7 \implies -5 > -15 + 7 \implies -5 > -8 \quad (\text{True}) \]

    So, the point (5, -5) satisfies both inequalities.

  2. Point (10, -8):

    • For the first inequality: \[ -8 < \frac{4}{5}(10) - 2 \implies -8 < 8 - 2 \implies -8 < 6 \quad (\text{True}) \]
    • For the second inequality: \[ -8 > -3(10) + 7 \implies -8 > -30 + 7 \implies -8 > -23 \quad (\text{True}) \]

    So, the point (10, -8) satisfies both inequalities.

  3. Point (5, -6):

    • For the first inequality: \[ -6 < \frac{4}{5}(5) - 2 \implies -6 < 4 - 2 \implies -6 < 2 \quad (\text{True}) \]
    • For the second inequality: \[ -6 > -3(5) + 7 \implies -6 > -15 + 7 \implies -6 > -8 \quad (\text{True}) \]

    So, the point (5, -6) satisfies both inequalities.

  4. Point (-10, -7):

    • For the first inequality: \[ -7 < \frac{4}{5}(-10) - 2 \implies -7 < -8 - 2 \implies -7 < -10 \quad (\text{False}) \]

    Since the first inequality is false, we don’t need to check the second inequality. Thus, the point (-10, -7) does not satisfy the first inequality.

Conclusion

So, the only point that does not satisfy the given system of inequalities is: (-10, -7).