Questions: Graph the system of linear inequalities below. Use your graph to determine which coordinate would be included in the solution to the system of linear inequalities -6x + 2y ≥ 4 8x - y ≤ -4 (0,2) (-2,1) (-1,-4) (5,0)

Solution Steps

To graph the system of inequalities, we first convert them to equations:

1. \( -6x + 2y = 4 \)
2. \( 8x - y = -4 \)

For the first equation \( -6x + 2y = 4 \):

• x-intercept: Set \( y = 0 \), then \( -6x = 4 \) which gives \( x = -\frac{2}{3} \).
• y-intercept: Set \( x = 0 \), then \( 2y = 4 \) which gives \( y = 2 \).

For the second equation \( 8x - y = -4 \):

• x-intercept: Set \( y = 0 \), then \( 8x = -4 \) which gives \( x = -\frac{1}{2} \).
• y-intercept: Set \( x = 0 \), then \( -y = -4 \) which gives \( y = 4 \).

Graph the lines and determine the solution region for the inequalities:

1. For \( -6x + 2y \geq 4 \), shade above the line.
2. For \( 8x - y \leq -4 \), shade below the line.

Test each point to see if it lies in the solution region:

• Point (0,2):

  • \( -6(0) + 2(2) = 4 \) (satisfies \( \geq 4 \))
  • \( 8(0) - 2 = -2 \) (does not satisfy \( \leq -4 \))

• Point (-2,1):

  • \( -6(-2) + 2(1) = 12 + 2 = 14 \) (satisfies \( \geq 4 \))
  • \( 8(-2) - 1 = -16 - 1 = -17 \) (satisfies \( \leq -4 \))

• Point (-1,-4):

  • \( -6(-1) + 2(-4) = 6 - 8 = -2 \) (does not satisfy \( \geq 4 \))
  • \( 8(-1) - (-4) = -8 + 4 = -4 \) (satisfies \( \leq -4 \))

• Point (5,0):

  • \( -6(5) + 2(0) = -30 \) (does not satisfy \( \geq 4 \))
  • \( 8(5) - 0 = 40 \) (does not satisfy \( \leq -4 \))

Final Answer