Inequalities and Symbols

Introduction

Key Concepts

1. Understanding Inequalities

An inequality is a mathematical statement that relates two expressions, indicating that one expression is larger or smaller than the other. Unlike equations, which assert that two expressions are equal, inequalities express a range of possible solutions. They are essential for describing relationships where values are not fixed but can vary within certain boundaries.

2. Inequality Symbols

Inequalities are represented using specific symbols that denote the relationship between two quantities. Understanding these symbols is fundamental to interpreting and solving inequality problems.

• Greater Than (>): Indicates that the value on the left is larger than the value on the right. For example, $5 > 3$ means five is greater than three.
• Less Than (<): Indicates that the value on the left is smaller than the value on the right. For example, $2 < 4$ means two is less than four.
• Greater Than or Equal To (≥): Indicates that the value on the left is either greater than or equal to the value on the right. For example, $x ≥ 7$ means $x$ is at least seven.
• Less Than or Equal To (≤): Indicates that the value on the left is either less than or equal to the value on the right. For example, $y ≤ 10$ means $y$ is at most ten.

3. Solving Inequalities

Solving inequalities involves finding all possible values of the variable that make the inequality true. The process is similar to solving equations but with additional considerations, especially when multiplying or dividing by negative numbers.

• Basic Steps:
  1. Isolate the variable on one side of the inequality.
  2. Perform operations to simplify the inequality.
  3. Remember to reverse the inequality symbol when multiplying or dividing by a negative number.
• Example: Solve $2x + 3 > 7$.
  1. Subtract 3 from both sides: $2x > 4$.
  2. Divide both sides by 2: $x > 2$.

  Solution: $x > 2$.

4. Graphing Inequalities on Number Lines

Graphing inequalities provides a visual representation of all possible solutions. On a number line, solutions to inequalities are depicted by shading the appropriate region and using open or closed circles to indicate whether the boundary is included.

• Types of Boundaries:
  • Open Circle: Represents that the boundary value is not included in the solution (used with > and <).
  • Closed Circle: Represents that the boundary value is included in the solution (used with ≥ and ≤).
• Example: Graph $x ≥ 5$ on a number line.
  • Draw a number line and place a closed circle at 5.
  • Shade the region to the right of 5, indicating all numbers greater than or equal to 5.

5. Types of Inequalities

Inequalities can be classified based on their structure and the relationships they describe. Understanding these types enhances the ability to solve complex problems.

• Linear Inequalities: Involve linear expressions. For example, $3x - 2 ≤ 7$.
• Quadratic Inequalities: Involve quadratic expressions. For example, $x^2 - 4 > 0$.
• Compound Inequalities: Combine two inequalities into one statement. For example, $1 < x ≤ 5$.

6. Compound Inequalities

Compound inequalities consist of two or more inequalities combined into a single statement. They are used to express that a variable lies within a certain range.

• Conjunctions: Use "and" to indicate that both conditions must be satisfied. For example, $2 < x < 6$ means $x$ is greater than 2 and less than 6.
• Disjunctions: Use "or" to indicate that at least one condition must be satisfied. For example, $x ≤ 1$ or $x ≥ 5$.

7. Solving Quadratic Inequalities

Quadratic inequalities involve expressions of degree two. Solving them requires identifying the critical points where the expression equals zero and testing intervals to determine where the inequality holds.

• Example: Solve $x^2 - 5x + 6 > 0$.
  1. Factor the quadratic: $(x - 2)(x - 3) > 0$.
  2. Find critical points: $x = 2$ and $x = 3$.
  3. Test intervals:
     • For $x < 2$, choose $x = 1$: $(1 - 2)(1 - 3) = ( -1)( -2) = 2 > 0$.
     • For $2 < x < 3$, choose $x = 2.5$: $(2.5 - 2)(2.5 - 3) = (0.5)( -0.5) = -0.25 < 0$.
     • For $x > 3$, choose $x = 4$: $(4 - 2)(4 - 3) = (2)(1) = 2 > 0$.
  4. Solution: $x < 2$ or $x > 3$.

8. Absolute Inequalities

Absolute inequalities involve absolute value expressions. Solving them requires considering both the positive and negative scenarios of the inner expression.

• Example: Solve $|x - 4| < 3$.
  1. Interpret the inequality: $-3 < x - 4 < 3$.
  2. Add 4 to all parts: $1 < x < 7$.

  Solution: $1 < x < 7$.

9. Systems of Inequalities

Systems of inequalities consist of multiple inequalities that must be satisfied simultaneously. Solving them involves finding the intersection of their solution sets.

• Example: Solve the system:
  • $y ≥ 2x + 1$
  • $y ≤ -x + 5$
  1. Graph both inequalities on the same coordinate plane.
  2. Identify the region where both shaded areas overlap.
  3. The overlapping region represents all solution pairs $(x, y)$ that satisfy both inequalities.

10. Applications of Inequalities

Inequalities are widely used in various real-life contexts, including finance, engineering, and everyday problem-solving, to model situations where constraints or limits are present.

• Budgeting: Determining spending limits, e.g., $Total\, Expenses ≤ Budget$.
• Construction: Ensuring materials meet strength criteria, e.g., $Stress < Maximum\, Stress\, Capacity$.
• Health: Setting dietary guidelines, e.g., $Calories\, Consumed ≤ Daily\, Caloric\, Needs$.
• Transportation: Adhering to speed limits, e.g., $Speed ≤ 60\, km/h$.

Comparison Table

Summary and Key Takeaways