Math Problem Statement

Graph the inequality x ≤ 3.

Solution

To graph the inequality x3x \leq 3, we need to follow these steps:

Step-by-Step Instructions:

  1. Draw the Boundary Line:

    • The inequality x3x \leq 3 involves the vertical line at x=3x = 3.
    • Since the inequality includes \leq (less than or equal to), the boundary line should be solid because points on the line satisfy the inequality.
  2. Plot Points on the Boundary:

    • Identify points on the line where x=3x = 3. Examples of such points are (3,0)(3, 0), (3,2)(3, 2), and (3,2)(3, -2).
  3. Shade the Region:

    • The inequality x3x \leq 3 means all values of xx that are less than or equal to 3.
    • Shade the region to the left of the vertical line at x=3x = 3 because that represents the values where xx is less than or equal to 3.

Summary:

  • Solid vertical line at x=3x = 3.
  • Shade the region to the left of this line.

Would you like more details on this process or have any further questions?


Here are 5 related questions to expand this understanding:

  1. What happens if the inequality was x<3x < 3 instead of x3x \leq 3?
  2. How would you graph the inequality y2x+1y \geq 2x + 1?
  3. How can you determine whether to use a solid or dashed line for inequalities?
  4. What changes if the inequality involves both xx and yy, like y2x5y \leq 2x - 5?
  5. How do you handle graphing inequalities with horizontal lines, like y>4y > 4?

Tip: Always remember that a solid line means the boundary points are included (e.g., \leq or \geq), while a dashed line means they are not (e.g., << or >>).

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Math Problem Analysis

Mathematical Concepts

Inequalities
Coordinate Plane Graphing

Formulas

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Theorems

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Suitable Grade Level

Grades 6-8