Inequalities and Intervals

Introduction:

An inequality is a mathematical statement that indicates a relationship between two expressions that are not equal. Unlike equations, which assert that two expressions are exactly equal, inequalities show that one expression is either less than, greater than, less than or equal to, or greater than or equal to another expression. Inequalities are essential in mathematics as they are used to describe a range of possible values rather than a specific value, which has applications in various fields such as economics, engineering, and optimization problems.

Graphical Representation:

Inequalities can be visually represented on a number line or in a coordinate plane. On a number line, the solution set is typically shown as a shaded region, with open or closed circles indicating whether the boundary points are included or excluded. In the coordinate plane, inequalities can define half-planes, with the boundary line being dashed for strict inequalities (e.g., y > mx + c) and solid for non-strict inequalities (e.g., y ≥ mx + c).

Types of Inequalities

  1. Strict Inequalities
    • Less Than (<):
      • Example:
        x < 5
      • Graphical Representation:
        A number line where all values to the left of 5 are shaded, but 5 itself is not included (open circle on 5).
    • Greater Than (>)
      • Example:
        y > -2
      • Graphical Representation:
        A number line where all values to the right of -2 are shaded, but -2 itself is not included (open circle on -2).
  2. Non-Strict Inequalities
    • Less Than or Equal To (≤)
      • Example:
        x ≤ 7
      • Graphical Representation:
         A number line where all values to the left of 7, including 7, are shaded (closed circle on 7).
    • Greater Than or Equal To (≥)
      • Example:
         z ≥ 3
      • Graphical Representation:
        A number line where all values to the right of 3, including 3, are shaded (closed circle on 3).
  3. Compound Inequalities
    • AND Inequality:
      • a < x < b
        Example: -2 < x < 8
      • a ≤ x < b
        Example: 3 ≤ x < 15
      • a < x ≤ b
        Example: -5 < x ≤ 6
      • a ≤ x ≤ b
        Example: 7 ≤ x ≤ 22
    • Graphical Representation:
       A number line showing the interval between 2 and 6, not cluding both 2 and 6 (open circle on 2, and on 6).
      • Example:
        2 < x ≤ 6
    • OR Inequality:
      • x < a   or   x > b
        Example:   x < 6   or   x > 25
      • x ≤ a   or   x > b
        Example:   x ≤ -5   or   x > 17
      • x < a   or   x ≥ b
        Example:   x < -10   or   x ≥ -2
      • x ≤ a   or   x ≥ b
        Example:   x ≤ -3   or   x ≥ 5
      • AExample:
        x < -1 or x > 4
      • Graphical Representation:
        A number line showing all values less than -1 and greater than 4 (open circles on -1 and 4).

Intervals

Introduction:

An interval in mathematics is a range of numbers between two endpoints. It represents all the numbers that lie within a certain range on the number line. Intervals are used to describe subsets of real numbers and can be either finite or infinite, depending on the endpoints. Intervals are crucial in various mathematical contexts, such as solving inequalities, analyzing functions, and determining domains and ranges.

Types of Intervals

  1. Closed Interval: [2, 5]
    • Definition:A closed interval includes both of its endpoints.
    • Notation: [a, b]
    • Example:[2, 5]
      • This interval includes all the numbers between 2 and 5, including 2 and 5 themselves.
    • Graphical Representation:
      • A number line with a shaded region between 2 and 5, with solid dots (closed circles) on 2 and 5.
  2. Open Interval: (3, 7)
    • Definition:An open interval excludes both of its endpoints.
    • Notation: (a, b)
    • Example:(3, 7)
      • This interval includes all the numbers between 3 and 7, but not 3 and 7 themselves.
    • Graphical Representation:
      • A number line with a shaded region between 3 and 7, with open circles on 3 and 7.
  3. Half-Open (or Half-Closed) Interval:
    • Definition:A half-open interval includes one endpoint but excludes the other.
    • Type:
      • Left-closed, right-open: [a, b)
      • Left-open, right-closed: (a, b]
    • Examples:
      • Left-closed, right-open: [4,9) includes 4 but excludes 9.
      • Left-open, right-closed: (2,6] excludes 2 but includes 6.
    • Graphical Representation:
      • Left-closed, right-open: Solid dot on 4, open circle on 9.
      • Left-open, right-closed: Open circle on 2, solid dot on 6.
  4. Infinite Interval:
    • Definition:An infinite interval extends indefinitely in one or both directions.
    • Type:
      • Open to the right: [a,∞)
      • Open to the left: (- ∞, b]
      • Infinite on both sides: (- ∞, ∞)
    • Example:
      • Open to the right: (5, ∞) includes all numbers greater than 5.
      • Open to the left: (- ∞, -3] includes all the numbers less than or equal to -3.
    • Graphical Representation:
      • Open to the right:Shaded region starting from 5 extending indefinitely to the right with an open circle on 5.
      • Open to the left: Shaded region extending indefinitely to the left up to -3 with a solid dot on -3.

Summary

Intervals are a fundamental concept for expressing ranges of numbers, especially in the context of inequalities and functions. By understanding the different types of intervals—closed, open, half-open, and infinite—one can accurately describe and analyze the subsets of the real number line that satisfy specific conditions.