Author Jonathan David
Author Jonathan David
College Algebra

Equations and Inequalities

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Equations and Inequalities

Key Terms and Definitions

  • Equation: A mathematical statement that asserts the equality of two expressions, often written in the form ax + b = c.
  • Inequality: A mathematical statement that compares two expressions using symbols such as >, <, \geq, or \leq.
  • Solution: The value or range of values that satisfies an equation or inequality.
  • Boundary: In inequalities, the point(s) where the inequality transitions from true to false.

Worked Example: Solve 3x + 4 = 10 and 2x - 5 \leq 7

  1. Solving the Equation:
    • Step 1: Subtract 4 from both sides:
      3x + 4 - 4 = 10 - 4
      3x = 6
    • Step 2: Divide by 3:
      \frac{3x}{3} = \frac{6}{3}
      x = 2
    • Verification: Substitute x = 2 into the original equation:
      3(2) + 4 = 10
      6 + 4 = 10, which is correct.
  2. Solving the Inequality:
    • Step 1: Add 5 to both sides:
      2x - 5 + 5 \leq 7 + 5
      2x \leq 12
    • Step 2: Divide by 2:
      \frac{2x}{2} \leq \frac{12}{2}
      x \leq 6
    • Solution: x \leq 6, meaning x can be any value less than or equal to 6.

Tips and Tricks

  • For equations, always perform operations equally on both sides to maintain balance.
  • For inequalities, remember to reverse the inequality sign if multiplying or dividing by a negative number.
  • Graphing inequalities can help visualize solutions.

Test-Taking Strategies

  • Double-check your operations, especially when dealing with negative numbers.
  • For compound inequalities, solve each part individually and then combine the results.
  • Clearly state your solution, using proper notation for inequalities (e.g., interval notation).

Understanding equations and inequalities is crucial for solving real-world problems, analyzing relationships, and preparing for advanced topics like systems of equations and optimization. Mastery of these concepts strengthens your mathematical foundation and enhances your problem-solving skills.

“It is not knowledge, but the act of learning, that grants the greatest enjoyment.” – Carl Friedrich Gauss

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