Section: Solving Linear and Rational Equations

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one. Solving these equations involves isolating the variable on one side of the equation.

• Definition: A linear equation in one variable has the general form , where and are constants.

• Steps to Solve:

  1. Expand both sides if necessary.

  2. Combine like terms.

  3. Isolate the variable by performing inverse operations.

  4. Solve for the variable.

• Example: Solve

  • Expand:

  • Subtract from both sides:

  • Subtract $22

  • Solution:

Solving Equations with Absolute Value

Equations involving absolute value require considering both the positive and negative scenarios for the expression inside the absolute value.

• Definition: The absolute value of , written , is the distance from to $0$ on the number line.

• General Rule: implies or .

• Example: Solve

  • Set up two equations: or

  • Solve: or

Solving Equations with Distribution and Combining Like Terms

Some linear equations require distributing and combining like terms before isolating the variable.

• Example: Solve

  • Combine like terms:

  • Equation:

  • Multiply both sides by 2 to clear the fraction:

  • Subtract from both sides:

  • Solve:

Solving Rational Equations

Rational equations contain variables in the denominator. To solve, find a common denominator and multiply both sides to clear fractions.

• Definition: A rational equation is an equation involving at least one rational expression, which is a ratio of polynomials.

• Steps to Solve:

  1. Identify the least common denominator (LCD).

  2. Multiply both sides by the LCD to eliminate denominators.

  3. Solve the resulting equation.

  4. Check for extraneous solutions by substituting back into the original equation.

• Example: Solve

  • Factor denominator:

  • LCD is

  • Multiply both sides by :

  • Expand:

  • Combine like terms:

  • Solve:

  • Check: Ensure (would make denominator zero). is valid.

Summary Table: Types of Equations and Solution Methods

Additional info: The notes are based on typical Precalculus algebraic equation-solving techniques, including linear, absolute value, and rational equations. The examples and steps have been expanded for clarity and completeness.