Linear and Absolute Value Equations and Inequalities

Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. Solving linear equations is a foundational skill in algebra and precalculus.

• Definition: A linear equation is an equation of the form , where , , and are constants.

• Steps to Solve:

  1. Simplify both sides by removing grouping symbols and combining like terms.

  2. Isolate the variable term on one side of the equation.

  3. Solve for the variable by performing inverse operations.

• Example: Solve Add 3 to both sides: Divide by 2:

Linear Inequalities

Linear inequalities are similar to linear equations but use inequality symbols (<, >, ≤, ≥) instead of an equal sign. The solution is often represented as an interval or graphed on a number line.

• Definition: A linear inequality is an inequality of the form , , , etc.

• Key Rule: When multiplying or dividing both sides by a negative number, reverse the direction of the inequality symbol.

• Example: Solve Add 4: Divide by 2: Interval notation:

Equations Involving Absolute Value

Absolute value equations involve expressions within absolute value bars, which represent the distance from zero on the number line. The solution process often involves splitting the equation into two cases.

• Definition: The absolute value of , written , is the non-negative value of .

• Solving Steps:

  1. Isolate the absolute value expression.

  2. Set up two equations: one for the positive case and one for the negative case.

  3. Solve each equation separately.

• Example: Solve Case 1: Case 2: Solution: or

Absolute Value Inequalities

Absolute value inequalities require careful setup, often resulting in compound inequalities. The solution set may be a range of values or two separate intervals.

• Types:

  • leads to (AND inequality)

  • leads to or (OR inequality)

• Example: Solve Add 2: Interval notation:

Compound Inequalities

Compound inequalities combine two inequalities using "and" or "or". The solution set is the intersection (AND) or union (OR) of the individual solution sets.

• AND: Both conditions must be true (overlap of solution sets).

• OR: At least one condition must be true (combine solution sets).

• Example: Solution:

Special Cases of Absolute Value Inequalities

Some absolute value inequalities have no solution or are true for all real numbers, depending on the values involved.

• No Solution: (since absolute value is always non-negative)

• All Real Numbers: (always true for any )

Table: Absolute Value Inequality Types

Graphing Solutions

Solutions to inequalities and absolute value equations are often represented on a number line. Use open circles for strict inequalities (<, >) and closed circles for inclusive inequalities (≤, ≥). Shade the region corresponding to the solution set.

• Example: is graphed with an open circle at 2 and shading to the right.

• Example: is graphed with closed circles at -3 and 4, shading between them.

Additional info:

• These notes cover topics from Ch. P (Prerequisites: Fundamental Concepts of Algebra) and Ch. 1 (Functions and Graphs) as they relate to equations and inequalities, which are foundational for all subsequent precalculus topics.