Graphing Intervals and Inequalities

Graphing Intervals

Examples of Graphing Sets

  1. Graph the following sets:

    • a) (1, 3) [2, 7]
      • Explanation: The first part (1, 3) represents an open interval where 1 and 3 are not included, while [2, 7] represents a closed interval where both 2 and 7 are included. The graph would depict two intervals on a number line, highlighting the open nature of (1, 3) with open circles at 1 and 3 and closed circles at 2 and 7.
    • b) (1, 3)[2, 7]
      • Explanation: This appears to represent similar intervals as in (a); however, it's noted that there might be no space between the two sets in this layout. Graphing would show the same as in (a).
    • c) [-4, 6][0, 8)
      • Explanation: Here, the intervals are broken down as follows: [-4, 6] is a closed interval including both -4 and 6, while [0, 8) is a mixed interval including 0 but excluding 8. The graph would show the closed circles at -4 and 6 and an open circle at 8. The section between -4 to 6 will be shaded in.
    • d) (-2, 0) (-1, 1)
      • Explanation: In this particular case with two separate intervals, the interval (-2, 0) is open, excluding both endpoints, and (-1, 1) is also open. Each would be graphed as open circles on the number line, with shading between -2 and 0 and between -1 and 1.

  2. Express each interval in terms of inequalities, and then graph the interval:

    • a) [-1, 2)
      • Inequality Representation: (-1 \leq x < 2)
      • Graphing: The graph would show a closed circle on -1 indicating it is included, and an open circle on 2 indicating it is excluded. The area between these two points would be shaded in.
    • b) [1.5, 4]
      • Inequality Representation: (1.5 \leq x \leq 4)
      • Graphing: The graph will have closed circles on both 1.5 and 4, indicating that both endpoints are included. The area between these points will be shaded in as well.

Summary of Key Concepts

  • Open Interval (a, b): The values within the interval from a to b but not including a and b themselves.
  • Closed Interval [a, b]: The values within the interval from a to b including both endpoints a and b.
  • Graphing Intervals: Utilize a number line for visual representation, indicating included or excluded values with closed/open circles accordingly.