Equations and Inequalities

Linear Equations and Rational Equations

This section introduces the fundamental concepts and methods for solving linear equations and rational equations in one variable. It also covers the classification of equations and the application of mathematical models to real-world problems.

Definition of a Linear Equation

• Linear Equation: An equation in one variable x that can be written in the form , where a and b are real numbers and a \neq 0.

• Linear equations are the foundation for solving more complex algebraic problems.

Generating Equivalent Equations

To solve equations, we often transform them into equivalent forms using the following operations:

• Remove grouping symbols and combine like terms.

• Add or subtract the same real number or variable expression on both sides.

• Multiply or divide both sides by the same nonzero quantity.

• Interchange the two sides of the equation.

Solving a Linear Equation

Follow these steps to solve a linear equation:

1. Simplify each side by removing grouping symbols and combining like terms.

2. Collect all variable terms on one side and constant terms on the other.

3. Isolate the variable and solve for its value.

4. Check the proposed solution in the original equation.

Example: Solving a Linear Equation

• Example: Solve .

  • Subtract from both sides:

  • Simplify:

  • Add $7x = 12$

  • Check:

Solving Linear Equations Involving Fractions

• To solve equations with fractions, multiply both sides by the least common denominator (LCD) to clear the fractions.

• Example: Solve

  • LCD is 28. Multiply both sides by 28:

  • Simplify:

  • Divide both sides by 12:

Solving Rational Equations

A rational equation contains at least one variable in the denominator. Solving such equations requires careful attention to restrictions on the variable.

• Set each denominator equal to zero to find restricted values (values that make the denominator zero).

• Multiply both sides by the LCD to clear denominators.

• Solve the resulting equation and check for extraneous solutions.

• Example: Solve

  • Restricted values: and

  • LCD is

  • Multiply both sides by LCD and solve for

Types of Equations: Identity, Conditional, Inconsistent

• Conditional Equation: True for at least one real number (e.g., ).

• Identity: True for all real numbers (e.g., ).

• Inconsistent Equation: Not true for any real number (e.g., ).

Example: Categorizing an Equation

• Inconsistent Equation: is false for all ; no solution exists.

• Identity: is true for all ; solution set is all real numbers.

Solving Applied Problems Using Mathematical Models

Mathematical models use equations to represent real-world situations. For example, the relationship between depression level and intensity of negative life events can be modeled by an equation.

• Example: If , and the low humor group averages a depression level of 10, solve for .

  • Set :

  • Divide both sides by 2.7: