Fundamentals - Examples & Applications

This section provides comprehensive, practical examples and step-by-step solutions to apply the foundational concepts of algebra. This includes classifying numbers, applying properties of equality and operations, evaluating absolute values, and solving simple and compound inequalities.

Sets of Numbers

Understanding the different sets within the real number system is critical for determining domains, ranges, and valid solutions to algebraic equations. Here are practical case studies classifying numbers.

Example

Case Study 1: Classifying Basic Real Numbers Classify the following numbers into all applicable sets (Natural N\mathbb{N}, Whole W\mathbb{W}, Integer Z\mathbb{Z}, Rational Q\mathbb{Q}, Irrational I\mathbb{I}, Real R\mathbb{R}):
  1. 77
  2. −3-3
  3. 0.250.25
  4. 5\sqrt{5}

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Example

Case Study 2: Classifying Complex Expressions Simplify and classify the following expressions into the most specific number set possible:
  1. 123\frac{12}{3}
  2. 16\sqrt{16}
  3. π+1\pi + 1
  4. 05\frac{0}{5}

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Prime Factorization, GCD, and LCM

Understanding the building blocks of numbers is crucial for simplifying fractions and solving problems involving repeating events.

Example

Example 1: Prime Factorization (Basic) Find the prime factorization of 6060.

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Example

Example 2: Greatest Common Divisor (Intermediate) Find the GCD of 4848 and 180180.

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Example

Example 3: Least Common Multiple (Advanced) Find the LCM of 2424, 3636, and 4040.

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Order of Operations (PEMDAS)

The order of operations guarantees that mathematical expressions are evaluated consistently. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division (left-to-right), and Addition and Subtraction (left-to-right).

Example

Example 1: Basic Application of PEMDAS Evaluate the expression: 15−3×2+8÷415 - 3 \times 2 + 8 \div 4

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Example

Example 2: Intermediate Application with Parentheses and Exponents Evaluate the expression: 42−2(5+3×2)+104^2 - 2(5 + 3 \times 2) + 10

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Example

Example 3: Advanced Application with Nested Grouping Symbols Evaluate the expression: 3[12−(4+22)]23−5\frac{3[12 - (4 + 2^2)]}{2^3 - 5}

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Common PEMDAS Error

A frequent mistake is interpreting "MD" and "AS" strictly sequentially. Multiplication does not take precedence over division; they are executed strictly from left-to-right. The same applies to addition and subtraction. For example, 10−4+210 - 4 + 2 is (10−4)+2=8(10 - 4) + 2 = 8, NOT 10−(4+2)=410 - (4 + 2) = 4.

Properties of Real Numbers

The Commutative, Associative, and Distributive properties are fundamental tools for manipulating algebraic expressions.

Example

Example 1: Identifying Properties Identify the property of real numbers illustrated by each equation:
  1. 4+(x+3)=(4+x)+34 + (x + 3) = (4 + x) + 3
  2. 7⋅(2y)=(7⋅2)y7 \cdot (2y) = (7 \cdot 2)y
  3. 5(a+2)=5a+105(a + 2) = 5a + 10
  4. x+y=y+xx + y = y + x

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Example

Example 2: Applying the Distributive Property (Basic) Simplify the expression: −3(2x−5)-3(2x - 5)

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Example

Example 3: Applying Properties to Simplify Expressions (Intermediate) Simplify the expression completely: 4(3x+2)−2(x−5)4(3x + 2) - 2(x - 5)

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Example

Example 4: Simplifying Nested Expressions (Advanced) Simplify the expression: 2x−3[x−4(x+1)]2x - 3[x - 4(x + 1)]

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Properties of Equality

The properties of equality justify the valid transformations we apply to equations to isolate a variable.

Example

Case Study 1: Identifying Equality Properties State the property of equality that justifies each logical step:
  1. If x=5x = 5, then 5=x5 = x.
  2. If a=ba = b and b=7b = 7, then a=7a = 7.
  3. y+2=y+2y + 2 = y + 2
  4. If x=3x = 3, evaluate 2x+42x + 4. The step 2(3)+42(3) + 4 relies on which property?

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Example

Case Study 2: Constructing a Logical Proof Given that 2a+b=c2a + b = c and b=d−ab = d - a, prove that a+d=ca + d = c using properties of equality.

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Absolute Value

Absolute value represents distance from zero on a number line. Absolute value equations typically split into two separate cases because a distance can be covered by moving in a positive or negative direction.

Example

Example 1: Basic Absolute Value Equation Solve the equation: ∣x−4∣=7|x - 4| = 7

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Example

Example 2: Isolating the Absolute Value Term (Intermediate) Solve the equation: 3∣2x+1∣−5=103|2x + 1| - 5 = 10

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Example

Example 3: No Solution / Extraneous Solutions (Advanced) Solve the equation: ∣3x−2∣+8=4|3x - 2| + 8 = 4

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Example

Example 4: Absolute Value Equal to an Expression (Edge Case) Solve the equation: ∣x−2∣=2x−10|x - 2| = 2x - 10

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Inequalities & Interval Notation

Solving inequalities is similar to solving equations, with the crucial addition of the "Golden Rule of Inequalities": you must flip the inequality symbol when multiplying or dividing by a negative number.

Golden Rule

Whenever you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must reverse. Failure to do so is the most common error in inequality problems.

Example

Example 1: Basic Linear Inequality Solve the inequality and express the answer in interval notation: 4x−7>94x - 7 > 9

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Example

Example 2: Negative Coefficient (Intermediate) Solve the inequality and express the answer in interval notation: −3(x−2)≥15-3(x - 2) \ge 15

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Example

Example 3: Compound Inequality - "AND" (Advanced) Solve the compound inequality: −8<2x+4≤10-8 < 2x + 4 \le 10

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Example

Example 4: Compound Inequality - "OR" (Edge Case) Solve the compound inequality: 3x−1≤−103x - 1 \le -10 OR 5x+2>175x + 2 > 17

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