Ask a New Question
Menu

Graphing Linear Inequalities

1 answer

Graphing linear inequalities involves a few straightforward steps. Here’s a guide to help you understand how to graph linear inequalities.

Steps to Graph a Linear Inequality

  1. Convert the Inequality to an Equation: Start by converting the inequality to an equation. For example, if you have the inequality \( y < 2x + 3 \), convert it to the equation: \[ y = 2x + 3 \]

  2. Graph the Boundary Line:

    • Identify the Type of Inequality:
      • If the inequality is \( < \) or \( > \), use a dashed line to indicate that points on the line are not included in the solution set.
      • If the inequality is \( \leq \) or \( \geq \), use a solid line to indicate that points on the line are included in the solution set.
    • Find two points on the line (if necessary):
      • For example, you can find the y-intercept (where \( x = 0 \)) and another point (by choosing a value for \( x \)) to create a straight line.

  3. Choose a Test Point: Select a test point that is not on the boundary line. A common choice is the origin \((0,0)\), unless it lies on the boundary line. Substitute the coordinates of this point into the original inequality to determine if it satisfies the inequality.

  4. Shade the Appropriate Region:

    • If the test point satisfies the inequality, shade the region of the graph that includes the test point.
    • If the test point does not satisfy the inequality, shade the opposite region.

Example

Let’s graph the inequality \( y \leq -\frac{1}{2}x + 4 \).

  1. Convert to Equation: \[ y = -\frac{1}{2}x + 4 \]

  2. Graph the Boundary Line:

    • This is a \( \leq \) inequality, so we use a solid line.
    • Find the y-intercept: When \( x = 0 \), \( y = 4 \) (point (0,4)).
    • Find another point: When \( x = 2 \), \( y = -\frac{1}{2}(2) + 4 = 3 \) (point (2,3)).
    • Plot (0, 4) and (2, 3) and draw a solid line through them.

  3. Choose a Test Point: Test the point (0,0): \[ 0 \leq -\frac{1}{2}(0) + 4 \implies 0 \leq 4 \text{ (True)} \]

  4. Shade the Region: Since (0,0) satisfies the inequality, shade the area below the line (where \( y \) values are less than or equal to the linear equation).

Summary

You have now graphed a linear inequality! Remember to label your axes and the line for clarity. If you have any specific inequalities you'd like help with or more examples, feel free to ask!