Questions: Graph the system of linear inequalities in two variables. 4x + 2y ≤ 6 4x - 2y ≥ -18 x - 2y ≤ 6

Graph the system of linear inequalities in two variables. 4x + 2y ≤ 6 4x - 2y ≥ -18 x - 2y ≤ 6
Transcript text: Graph the system of linear inequalities in two variables. \[ \begin{array}{l} 4 x+2 y \leq 6 \\ 4 x-2 y \geq-18 \\ x-2 y \leq 6 \end{array} \]
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Solution

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Solution Steps

Step 1: Convert Inequalities to Equations

To graph the system of inequalities, we first convert each inequality into an equation by replacing the inequality sign with an equality sign.

  1. \(4x + 2y = 6\)
  2. \(4x - 2y = -18\)
  3. \(x - 2y = 6\)
Step 2: Solve for y in Each Equation

Solve each equation for \(y\) in terms of \(x\).

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

    \[ 2y = -4x + 6 \implies y = -2x + 3 \]

  2. \(4x - 2y = -18\)

    \[ -2y = -4x - 18 \implies y = 2x + 9 \]

  3. \(x - 2y = 6\)

    \[ -2y = -x + 6 \implies y = \frac{1}{2}x - 3 \]

Step 3: Determine the Inequality Regions

For each inequality, determine the region that satisfies the inequality:

  1. \(4x + 2y \leq 6\) corresponds to the region below or on the line \(y = -2x + 3\).
  2. \(4x - 2y \geq -18\) corresponds to the region above or on the line \(y = 2x + 9\).
  3. \(x - 2y \leq 6\) corresponds to the region below or on the line \(y = \frac{1}{2}x - 3\).

Final Answer

The system of inequalities is represented by the lines:

  1. \(y = -2x + 3\)
  2. \(y = 2x + 9\)
  3. \(y = \frac{1}{2}x - 3\)

The solution is the intersection of the regions defined by these inequalities.

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = -2x + 3", "y = 2x + 9", "y = (1/2)x - 3"], "latex_expressions": ["$y = -2x + 3$", "$y = 2x + 9$", "$y = \\frac{1}{2}x - 3$"]}

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