The Main Idea

• Definition of Rational Equations:
  • Contains at least one rational expression
  • Variable appears in at least one denominator
• Solving Process:
  • Factor all denominators
  • Find and exclude values that make denominators zero
  • Determine the Least Common Denominator (LCD)
  • Multiply both sides by the LCD
  • Solve the resulting equation
  • Check solutions in the original equation
• Cross-Multiplication Method:
  • For equations in the form [latex]\frac{a}{b} = \frac{c}{d}[/latex]
  • Multiply to get [latex]ad = bc[/latex]
• Handling Binomial Denominators:
  • Treat binomials (e.g., [latex]x + 1[/latex]) as single units
  • Factor completely before finding LCD
• Importance in Algebra:
  • Bridge between linear equations and more complex algebraic structures
  • Foundation for solving many real-world problems 

The Main Idea

• Definition:
  • Equations containing variables under a radical symbol
  • Example: [latex]\sqrt{3x + 18} = x[/latex]
• Solving Process:
  • Isolate the radical term
  • Raise both sides to the power of the radical’s index
  • Solve the resulting equation
  • Check for extraneous solutions
• Extraneous Solutions:
  • Solutions that satisfy the altered equation but not the original
  • Result from squaring or cubing both sides
  • Must be checked by substitution in the original equation
• Multiple Radicals:
  • Isolate one radical at a time
  • Repeat the process for each radical

Problem-Solving Strategy

1. Identify all radical terms
2. Plan the isolation sequence for multiple radicals
3. Raise both sides to appropriate powers
4. Solve the resulting polynomial equation
5. Check all solutions in the original equation

The Main Idea

• Rational Exponents:
  • Fractions as exponents
  • Notation: [latex]a^{\frac{m}{n}} = \sqrt[n]{a^m}[/latex]
• Equivalence to Radicals:
  • [latex]a^{\frac{1}{n}} = \sqrt[n]{a}[/latex]
  • [latex]a^{\frac{m}{n}} = (\sqrt[n]{a})^m[/latex]
• Solving Strategy:
  • Raise both sides to the reciprocal power
  • Simplify using exponent rules
  • Solve the resulting equation
• Key Exponent Rules:
  • Product: [latex]a^m \cdot a^n = a^{m+n}[/latex]
  • Quotient: [latex]\frac{a^m}{a^n} = a^{m-n}[/latex]
  • Power: [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]

Problem-Solving Technique

1. Identify the rational exponent
2. Determine its reciprocal
3. Apply the reciprocal exponent to both sides
4. Simplify using exponent rules
5. Solve the resulting equation
6. Check the solution

The Main Idea

• Definition: A polynomial equation is an equation of the form: [latex]a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0 = 0[/latex] where [latex]n[/latex] is a positive integer and [latex]a_n \neq 0[/latex]
• Degree:
  • The highest power of the variable in the polynomial
  • Determines the maximum number of solutions
• Zero-Product Property: If [latex]ab = 0[/latex], then [latex]a = 0[/latex] or [latex]b = 0[/latex]
  • Fundamental to solving polynomial equations by factoring
• Solution Types:
  • Real solutions (rational or irrational)
  • Complex solutions (when real solutions don’t exist)

Problem-Solving Steps

1. Arrange the polynomial in standard form (descending powers)
2. Factor out the greatest common factor (GCF)
3. Look for special patterns or grouping opportunities
4. Factor completely
5. Apply the zero-product property
6. Solve the resulting linear equations
7. Check solutions in the original equation

The Main Idea

• Definition of Absolute Value:
  • Represents the distance of a number from zero on the number line
  • Always non-negative
  • Formally defined as: [latex]|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}[/latex]
• Absolute Value Equation:
  • An equation containing an absolute value expression
  • General form: [latex]|A| = B[/latex], where A is an expression and B is a non-negative number
• Properties of Absolute Value Equations:
  • If [latex]|A| = B[/latex], then [latex]A = B[/latex] or [latex]A = -B[/latex] when [latex]B \geq 0[/latex]
  • If [latex]B < 0[/latex], the equation [latex]|A| = B[/latex] has no solution
• Standard Form of Linear Absolute Value Equations:
  • [latex]|ax + b| = c[/latex], where [latex]a \neq 0[/latex] and [latex]c[/latex] is a real number
• Number of Solutions:
  • If [latex]c < 0[/latex]: No solution
  • If [latex]c = 0[/latex]: One solution
  • If [latex]c > 0[/latex]: Two solutions

Solving Process:

1. Isolate the absolute value expression on one side of the equation
2. Consider two cases: positive and negative
3. Solve each case as a linear equation
4. Check solutions in the original equation