Equations & Inequalities

Solving Linear Equations

Linear equations are fundamental in algebra and involve expressions where the variable has a degree of one. Solving these equations requires systematic application of algebraic operations to isolate the variable.

• Definition: A linear equation is an equation of the form ax + b = c, where x is the variable and a, b, c are constants.

• Steps to Solve:

  1. Distribute constants if necessary.

  2. Combine like terms on each side.

  3. Group terms with x and constants on opposite sides.

  4. Isolate x (solve for x).

  5. Check the solution by substituting x back into the original equation.

• Example: Solve 3x = 12.

  • Divide both sides by 3:

  • Solution:

Linear Equations with Fractions

When linear equations contain fractions, it is often helpful to clear the fractions by multiplying both sides by the least common denominator (LCD).

• Steps:

  1. Multiply both sides by the LCD to eliminate fractions.

  2. Proceed as with standard linear equations.

• Example: Solve .

  • Multiply both sides by 4:

  • Subtract 2:

Categorizing Linear Equations

Linear equations can be classified based on their solution sets:

• Conditional Equation: Has one solution.

• Identity: True for all real numbers (infinite solutions).

• Inconsistent Equation: Has no solution.

• Example:

  • is an identity.

  • is inconsistent.

Solving Rational Equations

Definition & Method

A rational equation contains a variable in the denominator. Solutions must not make any denominator zero.

• Steps:

  1. Determine restrictions by setting denominators equal to zero.

  2. Multiply both sides by the LCD to clear fractions.

  3. Solve the resulting linear equation.

  4. Check solutions against restrictions.

• Example: Solve with .

The Imaginary Unit and Complex Numbers

Square Roots of Negative Numbers

Square roots of negative numbers are not real. The imaginary unit i is defined as .

• Example:

• General Formula: (where b is positive)

Powers of i

Powers of i cycle every four:

• For higher powers, divide the exponent by 4 and use the remainder to determine the value.

• Example:

Complex Numbers

A complex number is of the form a + bi, where a is the real part and b is the imaginary part.

• Example: has real part 4 and imaginary part 3.

Operations with Complex Numbers

• Add/Subtract: Combine like terms.

• Multiply: Use distributive property and simplify .

• Complex Conjugate: The conjugate of is . Multiplying a complex number by its conjugate yields a real number.

• Divide: Multiply numerator and denominator by the conjugate of the denominator to rationalize.

Quadratic Equations

Standard Form

A quadratic equation is a polynomial of degree 2, typically written as .

• Example: can be rewritten as

Factoring Quadratic Equations

Factoring is one method to solve quadratic equations. Set each factor equal to zero and solve for x.

• Example: factors to

• Solutions: and

The Square Root Property

When a quadratic is in the form , take the square root of both sides to solve for x.

• Example: yields

• Solutions:

Completing the Square

Completing the square transforms a quadratic into form, allowing use of the square root property.

• Steps:

  1. Rewrite equation as

  2. Add to both sides

  3. Factor left side as a perfect square

  4. Solve using square root property

• Example: becomes

The Quadratic Formula

The quadratic formula solves any quadratic equation in standard form:

• Example: Solve :

  • Solutions: and

• 

The Discriminant

The discriminant, , determines the number and type of solutions for a quadratic equation.

• If : Two real solutions

• If : One real solution

• If : Two complex (imaginary) solutions

Linear Inequalities

Interval Notation

Interval notation is a compact way to express solution sets for inequalities. It uses parentheses for open intervals and brackets for closed intervals.

• Closed Interval: means

• Open Interval: means

• Half-Open Interval: or

• Infinity: Use or for unbounded intervals, always with parentheses.

• 

• 

Solving Linear Inequalities

Linear inequalities are solved similarly to linear equations, but the solution is a range of values. When multiplying or dividing by a negative number, reverse the inequality symbol.

• Example: Solve .

  • Subtract 12:

  • Divide by 2:

  • Interval notation:

• Example with negative: becomes after dividing by -2 and reversing the symbol.

Fractions & Variables on Both Sides

Inequalities with fractions or variables on both sides are solved by clearing fractions and isolating the variable, just as with equations.

• Example:

• Multiply both sides by 4:

• Subtract 2:

• Interval notation: