To graph the inequality \(x + 2y \leq -4\), follow these steps:
Step 1: Convert the Inequality to an Equation
Start by converting the inequality to an equation for easier graphing. The corresponding equation is: \[ x + 2y = -4 \]
Step 2: Find the Intercepts
To graph the line, find the x-intercept and y-intercept.
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X-intercept: Set \(y = 0\) in the equation: \[ x + 2(0) = -4 \implies x = -4 \] So, the x-intercept is \((-4, 0)\).
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Y-intercept: Set \(x = 0\) in the equation: \[ 0 + 2y = -4 \implies 2y = -4 \implies y = -2 \] So, the y-intercept is \((0, -2)\).
Step 3: Plot the Line
Plot the points \((-4, 0)\) and \((0, -2)\) on the coordinate plane. Next, draw a line through these points. Because the original inequality is \(\leq\), you will use a solid line to indicate that points on the line are included in the solution.
Step 4: Shade the Region
Since the inequality is \(x + 2y \leq -4\), you want to shade the region below the line. This is the region where the y-values satisfy the inequality.
Summary Coordinate Points
You can use the following coordinates that represent points that satisfy the inequality:
- A point below the line (for example, \((-5, 0)\)): \[ -5 + 2(0) = -5 \leq -4 \quad \text{(True)} \]
- Another point below the line (for example, \((-4, -3)\)): \[ -4 + 2(-3) = -4 - 6 = -10 \leq -4 \quad \text{(True)} \]
Final Note
You can pick many coordinates below the line that will satisfy the inequality. Just ensure that they satisfy \(x + 2y \leq -4\).
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