Question

Solution Sets of Simultaneous Inequalities Quick Check 1 of 51 of 5 Items Question 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.(1 point) Responses Yes, because both of these inequalities include points on the line as a solution. Yes, because both of these inequalities include points on the line as a solution. No, it is in the solution set for y≤6x−3, but on the line for y>x+5. No, it is in the solution set for y less-than-or-equal-to 6 x minus 3 , but on the line for y greater than x plus 5 . Yes, it is within the solution set for both inequalities. Yes, it is within the solution set for both inequalities. No, it is not in the solution set for either inequality. No, it is not in the solution set for either inequality. Skip to navigation

Answer 1

To determine whether the point (3, 8) is a solution to the simultaneous inequalities \(y > x + 5\) and \(y \leq 6x - 3\), we will evaluate each inequality with \(x = 3\) and \(y = 8\):

1. Check the first inequality:
   \[ y > x + 5 \] Substitute \(x = 3\) and \(y = 8\): \[ 8 > 3 + 5 \] \[ 8 > 8 \] This is false because \(8\) is not greater than \(8\).

2. Check the second inequality:
   \[ y \leq 6x - 3 \] Substitute \(x = 3\) and \(y = 8\): \[ 8 \leq 6(3) - 3 \] \[ 8 \leq 18 - 3 \] \[ 8 \leq 15 \] This is true because \(8\) is indeed less than or equal to \(15\).

Since the point (3, 8) satisfies the second inequality \(y \leq 6x - 3\) but does not satisfy the first inequality \(y > x + 5\), 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? (1 point) Responses (2,2) left parenthesis 2 comma 2 right parenthesis (10,5) left parenthesis 10 comma 5 right parenthesis (25,12) left parenthesis 25 comma 12 right parenthesis (3,6)

Answer 3

To determine which point is a solution to the simultaneous inequalities \( y < 12x \) and \( y > 13x + 3 \), we will evaluate each point against both inequalities:

1. For the point (2, 2):

   • \( y < 12x \)
     \( 2 < 12(2) \)
     \( 2 < 24 \) → True
   • \( y > 13x + 3 \)
     \( 2 > 13(2) + 3 \)
     \( 2 > 26 + 3 \)
     \( 2 > 29 \) → False
     (Not a solution)

2. For the point (10, 5):

   • \( y < 12x \)
     \( 5 < 12(10) \)
     \( 5 < 120 \) → True
   • \( y > 13x + 3 \)
     \( 5 > 13(10) + 3 \)
     \( 5 > 130 + 3 \)
     \( 5 > 133 \) → False
     (Not a solution)

3. For the point (25, 12):

   • \( y < 12x \)
     \( 12 < 12(25) \)
     \( 12 < 300 \) → True
   • \( y > 13x + 3 \)
     \( 12 > 13(25) + 3 \)
     \( 12 > 325 + 3 \)
     \( 12 > 328 \) → False
     (Not a solution)

4. For the point (3, 6):

   • \( y < 12x \)
     \( 6 < 12(3) \)
     \( 6 < 36 \) → True
   • \( y > 13x + 3 \)
     \( 6 > 13(3) + 3 \)
     \( 6 > 39 + 3 \)
     \( 6 > 42 \) → False
     (Not a solution)

Based on the evaluations, none of the points provided are solutions to the simultaneous inequalities \( y < 12x \) and \( y > 13x + 3 \).

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 (1,5.5) left parenthesis 1 comma 5.5 right parenthesis (−1.6,4.2) left parenthesis negative 1.6 comma 4.2 right parenthesis (4,8) left parenthesis 4 comma 8 right parenthesis (−5,5)

Answer 5

To determine which point is a solution to the simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \), we will evaluate each point against both inequalities.

1. For the point (1, 5.5):

   • Check \( y > \frac{1}{2}x + 5 \):
     \( 5.5 > \frac{1}{2}(1) + 5 \)
     \( 5.5 > 0.5 + 5 \)
     \( 5.5 > 5.5 \) → False
     (Fails this inequality)

2. For the point (-1.6, 4.2):

   • Check \( y > \frac{1}{2}x + 5 \):
     \( 4.2 > \frac{1}{2}(-1.6) + 5 \)
     \( 4.2 > -0.8 + 5 \)
     \( 4.2 > 4.2 \) → False
     (Fails this inequality)

3. For the point (4, 8):

   • Check \( y > \frac{1}{2}x + 5 \):
     \( 8 > \frac{1}{2}(4) + 5 \)
     \( 8 > 2 + 5 \)
     \( 8 > 7 \) → True
   • Check \( y < -2x + 1 \):
     \( 8 < -2(4) + 1 \)
     \( 8 < -8 + 1 \)
     \( 8 < -7 \) → False
     (Fails this inequality)

4. For the point (-5, 5):

   • Check \( y > \frac{1}{2}x + 5 \):
     \( 5 > \frac{1}{2}(-5) + 5 \)
     \( 5 > -2.5 + 5 \)
     \( 5 > 2.5 \) → True
   • Check \( y < -2x + 1 \):
     \( 5 < -2(-5) + 1 \)
     \( 5 < 10 + 1 \)
     \( 5 < 11 \) → True
     (This point satisfies both inequalities)

Based on the evaluations, the point (-5, 5) is a solution to the given simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \).

Answer 6

Select a true statement to describe the graphed system of inequalities. (1 point) Responses The system has infinite solutions. The system has infinite solutions. The system has two solutions. The system has two solutions. The system has one solution. The system has one solution. The system has no solutions. The system has no solutions. Skip to navigation page 17 of 17

Answer 7

To determine the correct statement about the graphed system of inequalities, we need to analyze how the lines representing the inequalities interact in the graph. However, since I cannot see the graph you're referring to, I'll provide you with guidelines on how to choose the right statement based on different scenarios.

1. Infinite Solutions: This is true if the shaded regions of the inequalities overlap in an area. This typically occurs when the two inequalities describe a region that has a common area where all points satisfy both inequalities.

2. Two Solutions: This would generally be inaccurate in the context of inequalities. Inequalities typically result in a range of solutions, not a finite number.

3. One Solution: This implies that the two inequalities intersect at exactly one point. However, this scenario is more common with equations rather than inequalities.

4. No Solutions: This is true if the shaded regions of the inequalities do not overlap at all. For example, if the regions described by the inequalities are completely separate, then there's no point that satisfies both inequalities.

Conclusion: If the graph shows overlapping shaded regions, select "The system has infinite solutions." If the graph shows entirely separate regions, select "The system has no solutions."

If you can describe the graph or provide specific information about it, I can help you choose a clearer answer!