Linear Equations and Inequalities in One Variable

Solving Linear Equations

Linear equations in one variable are fundamental in algebra and involve finding the value of the variable that makes the equation true. The process typically follows a systematic approach to isolate the variable.

• Simplify each side: Combine like terms and simplify expressions on both sides of the equation.

• Collect variable terms: Move all terms containing the variable to one side and constants to the other.

• Isolate the variable: Use inverse operations to solve for the variable.

• Check the solution: Substitute the solution back into the original equation to verify correctness.

Example: Solving a Linear Equation

Suppose we solve the equation :

• Simplify: Already simplified.

• Isolate: Subtract 2 from both sides: .

• Check: (True).

• Solution set: {6}

Solving Linear Equations with Fractions

Equations containing fractions can be simplified by clearing denominators using the least common denominator (LCD).

• Multiply both sides by the LCD: This eliminates fractions and simplifies the equation.

• Solve as usual: Proceed to isolate the variable.

Example: Solving with Fractions

Solve :

• LCD is 12. Multiply both sides by 12:

Solving Linear Equations with Decimals

To clear decimals, multiply every term by a power of 10 corresponding to the greatest number of decimal places.

• Multiply both sides: Use 10, 100, etc., to eliminate decimals.

• Solve: Proceed as with standard linear equations.

Example: Solving with Decimals

Solve :

• Distribute:

• Multiply both sides by 100:

Recognizing Inconsistent Equations and Identities

Some equations have no solution (inconsistent), while others are true for all real numbers (identities).

• Inconsistent equation: Results in a false statement (e.g., ). Solution set is empty ().

• Identity: Results in a true statement (e.g., ). Solution set is all real numbers.

Example: No Solution

Solve :

• (False)

• No solution:

Example: Identity

Solve :

• (True for all )

• Solution set: {x | x is a real number}

Solving Applied Problems Using Formulas

Linear equations are often used to model real-world situations, such as relationships between variables in scientific or social contexts.

• Identify variables: Assign variables to unknowns.

• Set up the equation: Use the given formula and substitute known values.

• Solve: Isolate the variable of interest.

Example: Application in Psychology

Suppose a formula models depression level in response to negative life event intensity . If , solve for :

• Substitute into the formula.

• Solve for to find the event intensity.

• Result:

Additional info: The included image is the cover of an Introductory Algebra textbook, directly relevant as it visually reinforces the academic context of the study notes.