Ask a New Question
Menu

Examine the following graphed system of linear inequalities.

A dashed line passing through the points (0, 2) and (6, 0) and a solid line passing through the points (0, 5), (-2, -9). The area to the bottom left of the intersecting point and between the lines is shaded.

Select the solution to the system of inequalities from the following points. Select all that apply.
A. (-5, 3)

B. (1, -2)

C.(0, -4)

D. (-8, -1)

E. (-3, 3)

1 answer

To determine which points satisfy the system of inequalities based on the description provided, we need to analyze the lines in the inequalities.

  1. Dashed line through (0, 2) and (6, 0): This represents a strict inequality. The equation of this line can be found using the slope-intercept form.

    The slope \(m\) is calculated as follows: \[ m = \frac{0 - 2}{6 - 0} = \frac{-2}{6} = -\frac{1}{3} \] So, the equation of the line is: \[ y = -\frac{1}{3}x + 2 \] Since it's a dashed line, the inequality for this line will be: \[ y < -\frac{1}{3}x + 2 \]

  2. Solid line through (0, 5) and (-2, -9): This indicates a non-strict inequality. The slope can be calculated as: \[ m = \frac{-9 - 5}{-2 - 0} = \frac{-14}{-2} = 7 \] The equation is: \[ y = 7x + 5 \] Since it's a solid line, the inequality for this line will be: \[ y \geq 7x + 5 \]

Now, we are looking for the shaded area that is below the dashed line and above the solid line. In other words, we need to find points that satisfy both:

  1. \(y < -\frac{1}{3}x + 2\)
  2. \(y \geq 7x + 5\)

Now let's evaluate the selected points.

A. (-5, 3)

  1. Substitute into \(y < -\frac{1}{3}(-5) + 2\): \[ 3 < \frac{5}{3} + 2 = \frac{5}{3} + \frac{6}{3} = \frac{11}{3} \quad \text{(True)} \]
  2. Substitute into \(y \geq 7(-5) + 5\): \[ 3 \geq -35 + 5 = -30 \quad \text{(True)} \]

B. (1, -2)

  1. Substitute into \(y < -\frac{1}{3}(1) + 2\): \[ -2 < -\frac{1}{3} + 2 = 2 - \frac{1}{3} = \frac{6}{3} - \frac{1}{3} = \frac{5}{3} \quad \text{(True)} \]
  2. Substitute into \(y \geq 7(1) + 5\): \[ -2 \geq 7 + 5 = 12 \quad \text{(False)} \]

C. (0, -4)

  1. Substitute into \(y < -\frac{1}{3}(0) + 2\): \[ -4 < 2 \quad \text{(True)} \]
  2. Substitute into \(y \geq 7(0) + 5\): \[ -4 \geq 5 \quad \text{(False)} \]

D. (-8, -1)

  1. Substitute into \(y < -\frac{1}{3}(-8) + 2\): \[ -1 < \frac{8}{3} + 2 = \frac{8}{3} + \frac{6}{3} = \frac{14}{3} \quad \text{(True)} \]
  2. Substitute into \(y \geq 7(-8) + 5\): \[ -1 \geq -56 + 5 = -51 \quad \text{(True)} \]

E. (-3, 3)

  1. Substitute into \(y < -\frac{1}{3}(-3) + 2\): \[ 3 < 1 + 2 = 3 \quad \text{(False)} \]
  2. Substitute into \(y \geq 7(-3) + 5\): \[ 3 \geq -21 + 5 = -16 \quad \text{(True)} \]

Now, compiling the results:

  • Point A: True
  • Point B: False
  • Point C: False
  • Point D: True
  • Point E: False

Final Selection: The points that satisfy the system of inequalities are A. (-5, 3) and D. (-8, -1).