Question

graph each system of inequalities and then answer the questions about the solution set of the system.
49.
y > -2
x + y ≥ 1
graph
is there a solution set? y n
is (0, 0) included in the solution set? y n
list one point included in the solution set that is not the origin.

50.
y ≥ 5 - x
y ≥ -x - 3
graph
is there a solution set? y n
is (0, 0) included in the solution set? y n
list one point included in the solution set that is not the origin.

Explanation:

Step1: Analyze Problem 49 (System: \( y > -2 \), \( x + y \geq 1 \))

• Is there a solution set?

The inequalities \( y > -2 \) (region above \( y = -2 \), dashed line) and \( x + y \geq 1 \) (region above/below \( x + y = 1 \), solid line) intersect. So, there is a solution set. Answer: Y.

• Is \( (0,0) \) included?

Check \( y > -2 \): \( 0 > -2 \) (true). Check \( x + y \geq 1 \): \( 0 + 0 = 0 \geq 1 \) (false). Thus, \( (0,0) \) is not included. Answer: N.

• List a non-origin point:

Find \( (x,y) \) where \( y > -2 \) and \( x + y \geq 1 \). Let \( x = 2 \), \( y = 0 \): \( 0 > -2 \) (true), \( 2 + 0 = 2 \geq 1 \) (true). So, \( (2,0) \) works.

Step2: Analyze Problem 50 (System: \( y \geq 5 - x \), \( y \geq -x - 3 \))

• Is there a solution set?

Both inequalities are "greater than or equal to" (solid lines). \( y \geq 5 - x \) (above line \( y = 5 - x \)) and \( y \geq -x - 3 \) (above line \( y = -x - 3 \)). These regions overlap (e.g., for large \( y \), both hold). So, there is a solution set. Answer: Y.

• Is \( (0,0) \) included?

Check \( y \geq 5 - x \): \( 0 \geq 5 - 0 \) (false, \( 0 \geq 5 \) is false). Thus, \( (0,0) \) is not included. Answer: N.

• List a non-origin point:

Find \( (x,y) \) where \( y \geq 5 - x \) and \( y \geq -x - 3 \). Let \( x = 0 \), \( y = 5 \): \( 5 \geq 5 - 0 \) (true), \( 5 \geq -0 - 3 \) (true). So, \( (0,5) \) works.

Answer:

Problem 49:

• Is there a solution set? \( \boldsymbol{Y} \)
• Is \( (0,0) \) included? \( \boldsymbol{N} \)
• Non-origin point: \( \boldsymbol{(2,0)} \) (or other valid point like \( (1,1) \))

Problem 50:

• Is there a solution set? \( \boldsymbol{Y} \)
• Is \( (0,0) \) included? \( \boldsymbol{N} \)
• Non-origin point: \( \boldsymbol{(0,5)} \) (or other valid point like \( (1,4) \))