Question

Graph the inequalities y<2x+2, y<2x−2, and y>−x. Which of the following ordered pairs are solutions to the set?(1 point)Responses(−1,−1), (2,−1), and (1,2)left parenthesis negative 1 comma negative 1 right parenthesis , left parenthesis 2 comma negative 1 right parenthesis , and left parenthesis 1 comma 2 right parenthesis(1.5,−1), (3,2), and (1,1)left parenthesis 1.5 comma negative 1 right parenthesis , left parenthesis 3 comma 2 right parenthesis , and left parenthesis 1 comma 1 right parenthesis(1.5,−1), (2,1), and (3,2)left parenthesis 1.5 comma negative 1 right parenthesis , left parenthesis 2 comma 1 right parenthesis , and left parenthesis 3 comma 2 right parenthesis(2,1), (−1,2), and (1,2)

Answer 1

To determine which ordered pairs are solutions to the set of inequalities \( y < 2x + 2 \), \( y < 2x - 2 \), and \( y > -x \), let's first analyze each inequality individually:

1. Inequality 1: \( y < 2x + 2 \)

   • This represents a region below the line \( y = 2x + 2 \).

2. Inequality 2: \( y < 2x - 2 \)

   • This corresponds to a region below the line \( y = 2x - 2 \).

3. Inequality 3: \( y > -x \)

   • This represents a region above the line \( y = -x \).

Next, we can sketch the appropriate graphs of these inequalities:

• The lines \( y = 2x + 2 \) and \( y = 2x - 2 \) are parallel lines with slopes equal to 2, and the second line is shifted down by 4 units. The area below both lines is where the first two inequalities are satisfied.
• The line \( y = -x \) has a slope of -1 and intersects the y-axis at \( (0, 0) \). The area above this line is where the third inequality is satisfied.

Now, let’s evaluate each of the listed ordered pairs:

1. \( (-1, -1) \):

   • For \( y < 2(-1) + 2 \): \( -1 < 0 \) (True)
   • For \( y < 2(-1) - 2 \): \( -1 < -4 \) (False)
   • For \( y > -(-1) \): \( -1 > 1 \) (False)
   • Not a solution.

2. \( (2, -1) \):

   • For \( y < 2(2) + 2 \): \( -1 < 6 \) (True)
   • For \( y < 2(2) - 2 \): \( -1 < 2 \) (True)
   • For \( y > -2 \): \( -1 > -2 \) (True)
   • This is a solution.

3. \( (1, 2) \):

   • For \( y < 2(1) + 2 \): \( 2 < 4 \) (True)
   • For \( y < 2(1) - 2 \): \( 2 < 0 \) (False)
   • For \( y > -1 \): \( 2 > -1 \) (True)
   • Not a solution.

4. \( (1.5, -1) \):

   • For \( y < 2(1.5) + 2 \): \( -1 < 5 \) (True)
   • For \( y < 2(1.5) - 2 \): \( -1 < 1 \) (True)
   • For \( y > -1.5 \): \( -1 > -1.5 \) (True)
   • This is a solution.

5. \( (3, 2) \):

   • For \( y < 2(3) + 2 \): \( 2 < 8 \) (True)
   • For \( y < 2(3) - 2 \): \( 2 < 4 \) (True)
   • For \( y > -3 \): \( 2 > -3 \) (True)
   • This is a solution.

6. \( (1, 1) \):

   • For \( y < 2(1) + 2 \): \( 1 < 4 \) (True)
   • For \( y < 2(1) - 2 \): \( 1 < 0 \) (False)
   • For \( y > -1 \): \( 1 > -1 \) (True)
   • Not a solution.

7. \( (2, 1) \):

   • For \( y < 2(2) + 2 \): \( 1 < 6 \) (True)
   • For \( y < 2(2) - 2 \): \( 1 < 2 \) (True)
   • For \( y > -2 \): \( 1 > -2 \) (True)
   • This is a solution.

8. \( (2, 1) \) was already tested, and it is a solution.

9. \( (-1, 2) \):

   • For \( y < 2(-1) + 2 \): \( 2 < 0 \) (False)
   • Not a solution.

10. \( (1, 2) \) was already tested as not a solution.

Summary of solutions: So the ordered pairs that are solutions to the set of inequalities are:

• \( (2, -1) \)
• \( (1.5, -1) \)
• \( (3, 2) \)
• \( (2, 1) \)

The valid response set would be \( (1.5, -1), (2, 1), (3, 2) \), as they all satisfy the inequalities.