Determine if the point (3,8) is a solution to the simultaneous inequalities y>x+5 and y≤6x−3. Select the response that gives the correct answer as well as the correct reasoning.

Question

Responses:

Yes, because both of these inequalities include points on the line as a solution.

No, it is not in the solution set for either inequality.

No, it is in the solution set for y≤6x−3, but on the line for y>x+5.

Yes, it is within the solution set for both inequalities.

Answer 1

To determine if the point (3,8) is a solution to the inequalities \( y > x + 5 \) and \( y \leq 6x - 3 \), we need to test both inequalities with the point.

1. Testing \( y > x + 5 \):

   • Substitute \( x = 3 \) and \( y = 8 \) into the inequality: \[ 8 > 3 + 5 \] \[ 8 > 8 \quad \text{(false)} \] Thus, the point (3,8) does not satisfy the first inequality.

2. Testing \( y \leq 6x - 3 \):

   • Substitute \( x = 3 \) and \( y = 8 \) into the inequality: \[ 8 \leq 6(3) - 3 \] \[ 8 \leq 18 - 3 \] \[ 8 \leq 15 \quad \text{(true)} \] Therefore, the point (3,8) satisfies the second inequality.

Since the point (3,8) does not satisfy the first inequality but does satisfy the second, the correct response is:

No, it is in the solution set for \( y \leq 6x - 3 \), but on the line for \( y > x + 5 \).

Answer 2

Which point is a solution to the simultaneous inequalities y<12x and y>13x+3 ?

Responses:
(2,2)

(3,6)

(25,12)

(10,5)

Answer 3

To find out which point satisfies the simultaneous inequalities \( y < 12x \) and \( y > 13x + 3 \), we will check each point one by one.

1. For the point (2, 2):

   • Check \( y < 12x \): \[ 2 < 12(2) \Rightarrow 2 < 24 \quad \text{(true)} \]
   • Check \( y > 13x + 3 \): \[ 2 > 13(2) + 3 \Rightarrow 2 > 26 + 3 \Rightarrow 2 > 29 \quad \text{(false)} \]
   • Conclusion: Not a solution.

2. For the point (3, 6):

   • Check \( y < 12x \): \[ 6 < 12(3) \Rightarrow 6 < 36 \quad \text{(true)} \]
   • Check \( y > 13x + 3 \): \[ 6 > 13(3) + 3 \Rightarrow 6 > 39 + 3 \Rightarrow 6 > 42 \quad \text{(false)} \]
   • Conclusion: Not a solution.

3. For the point (25, 12):

   • Check \( y < 12x \): \[ 12 < 12(25) \Rightarrow 12 < 300 \quad \text{(true)} \]
   • Check \( y > 13x + 3 \): \[ 12 > 13(25) + 3 \Rightarrow 12 > 325 + 3 \Rightarrow 12 > 328 \quad \text{(false)} \]
   • Conclusion: Not a solution.

4. For the point (10, 5):

   • Check \( y < 12x \): \[ 5 < 12(10) \Rightarrow 5 < 120 \quad \text{(true)} \]
   • Check \( y > 13x + 3 \): \[ 5 > 13(10) + 3 \Rightarrow 5 > 130 + 3 \Rightarrow 5 > 133 \quad \text{(false)} \]
   • Conclusion: Not a solution.

After checking all the points against the inequalities \( y < 12x \) and \( y > 13x + 3 \), none of the provided options satisfy both inequalities. Therefore, none of the given points is a solution to the simultaneous inequalities.

Answer 4

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)

(−5,5)

(4,8)

(1,5.5)

Answer 5

To determine which point satisfies the simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \), we will check each point one by one.

1. For the point \((-1.6, 4.2)\):

   • Check \( y > \frac{1}{2}x + 5 \): \[ 4.2 > \frac{1}{2}(-1.6) + 5 \Rightarrow 4.2 > -0.8 + 5 \Rightarrow 4.2 > 4.2 \quad \text{(false)} \]
   • Conclusion: Not a solution.

2. For the point \((-5, 5)\):

   • Check \( y > \frac{1}{2}x + 5 \): \[ 5 > \frac{1}{2}(-5) + 5 \Rightarrow 5 > -2.5 + 5 \Rightarrow 5 > 2.5 \quad \text{(true)} \]
   • Check \( y < -2x + 1 \): \[ 5 < -2(-5) + 1 \Rightarrow 5 < 10 + 1 \Rightarrow 5 < 11 \quad \text{(true)} \]
   • Conclusion: This point is a solution.

3. For the point \((4, 8)\):

   • Check \( y > \frac{1}{2}x + 5 \): \[ 8 > \frac{1}{2}(4) + 5 \Rightarrow 8 > 2 + 5 \Rightarrow 8 > 7 \quad \text{(true)} \]
   • Check \( y < -2x + 1 \): \[ 8 < -2(4) + 1 \Rightarrow 8 < -8 + 1 \Rightarrow 8 < -7 \quad \text{(false)} \]
   • Conclusion: Not a solution.

4. For the point \((1, 5.5)\):

   • Check \( y > \frac{1}{2}x + 5 \): \[ 5.5 > \frac{1}{2}(1) + 5 \Rightarrow 5.5 > 0.5 + 5 \Rightarrow 5.5 > 5.5 \quad \text{(false)} \]
   • Conclusion: Not a solution.

After testing all the points, the only point that satisfies both inequalities is:

\((-5, 5)\).