Equations & Inequalities

Solving 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. The general form is ax + b = c. Solving linear equations involves isolating the variable on one side of the equation using inverse operations.

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

• Steps to Solve:

  1. Distribute constants if necessary.

  2. Combine like terms on each side.

  3. Group terms with x on one side and constants on the other.

  4. Isolate x by performing inverse operations.

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

• Example: Solve

  • Distribute:

  • Add 6:

  • Divide by 2:

Linear Equations with Fractions

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

• Steps:

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

  2. Proceed as with standard linear equations.

• Example: Solve

  • LCD is 12. Multiply both sides by 12:

Categorizing Linear Equations

Linear equations can be classified based on the number of solutions:

• Conditional Equation: Has exactly one solution. Example:

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

• Inconsistent Equation: Has no solution. Example:

Solving Rational Equations

A rational equation contains at least one rational expression (a fraction with a variable in the denominator). To solve, clear denominators and check for extraneous solutions.

• Steps:

  1. Identify restrictions by setting denominators equal to zero.

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

  3. Solve the resulting linear equation.

  4. Check that solutions do not violate restrictions.

• Example: Solve

  • Restriction:

  • Cross-multiply:

  • Check: does not violate the restriction.

The Imaginary Unit and Complex Numbers

The Imaginary Unit

The imaginary unit, denoted as i, is defined as . It allows us to express square roots of negative numbers.

• Key Property:

• Example:

Powers of i

Powers of i repeat in a cycle of four:

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

• Example:

Complex Numbers

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

• Standard Form:

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

Operations with Complex Numbers

Adding and Subtracting

Combine like terms (real with real, imaginary with imaginary).

• Example:

Multiplying

Use distributive property or FOIL, and simplify using .

• Example:

Complex Conjugates

The conjugate of is . Multiplying a complex number by its conjugate yields a real number.

• Example:

Dividing Complex Numbers

To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator to make the denominator real.

• Example:

  • Multiply by :

  • Numerator:

  • Denominator:

  • Result:

Quadratic Equations

Introduction to Quadratic Equations

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

• Standard Form:

• Example:

Factoring Quadratic Equations

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

• Steps:

  1. Write the equation in standard form.

  2. Factor completely.

  3. Set each factor equal to zero.

  4. Solve for x.

• Example: factors to , so or .

The Square Root Property

If a quadratic equation can be written as , solve by taking the square root of both sides.

• Steps:

  1. Isolate the squared term.

  2. Take the positive and negative square roots of both sides.

  3. Solve for x.

• Example: gives , so or .

Completing the Square

Completing the square rewrites a quadratic in the form to solve for x.

• Steps:

  1. Move the constant to the other side:

  2. Add to both sides.

  3. Factor the left side as a perfect square trinomial.

  4. Solve using the square root property.

• Example: becomes , add $9x^2 + 6x + 9 = 8(x + 3)^2 = 8x + 3 = \pm \sqrt{8}$, $x = -3 \pm 2\sqrt{2}$.

The Quadratic Formula

The quadratic formula solves any quadratic equation :

• Example: becomes ; .

• Plug into the formula to find solutions.

The Discriminant

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

• If : Two distinct real solutions.

• If : One real solution (a repeated root).

• If : Two complex (imaginary) solutions.

Choosing a Method to Solve Quadratic Equations

There are four main methods to solve quadratic equations. The best method depends on the form of the equation:

• Factoring: Use if the equation factors easily.

• Square Root Property: Use if there is no linear term ().

• Completing the Square: Use if and is even.

• Quadratic Formula: Use if the equation does not factor easily or for any quadratic equation.

Linear Inequalities

Interval Notation

Interval notation is a concise way to describe sets of numbers, especially solution sets to inequalities.

• Closed Interval: includes both endpoints.

• Open Interval: excludes both endpoints.

• Half-Open Interval: or includes one endpoint.

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

• Example: is in interval notation.

Solving Linear Inequalities

Linear inequalities are similar to linear equations but use inequality symbols (<, >, ≤, ≥). The solution is often a range of values.

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

• Example: Solve :

  • Subtract 12:

  • Divide by 2:

  • Interval notation:

Inequalities with Fractions and Variables on Both Sides

Solve as you would a linear equation, being careful with the direction of the inequality when multiplying or dividing by negatives.

• Example:

  • Multiply both sides by 15:

  • (reverse the symbol)

  • Interval notation: