Solve each system of inequalities by graphing.

3x + 4y less then -14
x - 3y greater then or equal to 17

1 answer

To solve the system of inequalities by graphing, follow these steps:

  1. Rewrite the inequalities: We will rewrite the inequalities in slope-intercept form (y = mx + b), which makes them easier to graph.

    First inequality:
    \(3x + 4y < -14\)

    Solving for y: \[ 4y < -3x - 14 \] \[ y < -\frac{3}{4}x - \frac{14}{4} \] \[ y < -\frac{3}{4}x - \frac{7}{2} \]

    Second inequality:
    \(x - 3y \geq 17\)

    Solving for y: \[ -3y \geq -x + 17 \] \[ y \leq \frac{1}{3}x - \frac{17}{3} \]

  2. Graph the inequalities:

    Inequality 1: \(y < -\frac{3}{4}x - \frac{7}{2}\)

    • The boundary line is \(y = -\frac{3}{4}x - \frac{7}{2}\). This line will be dashed because the inequality is strict ("less than").
    • Find two points to graph the line. For example:
      • If \(x = 0\), then \(y = -\frac{7}{2} = -3.5\).
      • If \(x = 4\), then \(y = -\frac{3}{4}(4) - \frac{7}{2} = -3 - 3.5 = -6.5\).
    • Plot these points (0, -3.5) and (4, -6.5), and draw a dashed line through them. Shade the region below this line.

    Inequality 2: \(y \leq \frac{1}{3}x - \frac{17}{3}\)

    • The boundary line is \(y = \frac{1}{3}x - \frac{17}{3}\). This line will be solid because the inequality is inclusive ("greater than or equal to").
    • Find two points to graph the line. For example:
      • If \(x = 0\), then \(y = -\frac{17}{3} \approx -5.67\).
      • If \(x = 6\), then \(y = \frac{1}{3}(6) - \frac{17}{3} = 2 - \frac{17}{3} = 2 - 5.67 = -3.67\).
    • Plot these points (0, -5.67) and (6, -3.67), and draw a solid line through them. Shade the region below this line.
  3. Identify the solution region:

    • The solution to the system of inequalities is the intersection of the shaded areas from both inequalities. This is the region where both conditions are met.

Summary

  • The first inequality's region will be below the dashed line \(y = -\frac{3}{4}x - \frac{7}{2}\).
  • The second inequality's region will be below the solid line \(y = \frac{1}{3}x - \frac{17}{3}\).

Make sure to check a few points in the overlapping shaded region to confirm they satisfy both inequalities.