Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the form ax + b = c. The solution is the value of x that makes the equation true.

• Key Steps:

  1. Isolate the variable on one side of the equation.

  2. Simplify both sides as needed.

  3. Check your solution by substituting back into the original equation.

• Example: Solve

Solving Linear Inequalities

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

• Key Steps:

  1. Isolate the variable as with equations.

  2. If you multiply or divide by a negative number, reverse the inequality sign.

  3. Express the solution as an interval or set.

• Example: Solve

Complex Numbers

Definition and Operations

A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined by .

• Addition/Subtraction: Combine like terms:

• Multiplication: Use distributive property and :

• Example:

• Example:

Exponents and Radicals

Properties of Exponents

• Product Rule:

• Quotient Rule:

• Power Rule:

• Negative Exponent:

• Zero Exponent: (for )

Radicals and Rational Exponents

• Definition:

• Example:

• Operations: Use exponent rules for expressions with rational exponents.

Quadratic Equations

Solving by Factoring

• Set the equation to zero:

• Factor the quadratic expression.

• Set each factor equal to zero and solve for x.

• Example: or

Completing the Square

• Rewrite as (if ).

• Add to both sides to complete the square.

• Factor and solve for x.

• Example: or

Quadratic Formula

• For , the solutions are:

• Discriminant: determines the nature of the roots:

  • If : Two real solutions

  • If : One real solution

  • If : Two complex solutions

• Example: or

Distance and the Pythagorean Theorem

Distance Formula

The distance between two points and is:

• Example: Between and :

Pythagorean Theorem

• For a right triangle with legs a and b and hypotenuse c:

• Used to check if three points form a right triangle by comparing the squares of the side lengths.

Circles in the Coordinate Plane

Equation of a Circle

• The standard form is , where is the center and r is the radius.

• To find the equation:

  1. Identify the center and radius r.

  2. Substitute into the standard form.

• Example: Center , radius $5(x - 2)^2 + (y + 3)^2 = 25$

Finding Center and Radius from Equation

• Given , the center is and the radius is .

• If the equation is expanded, complete the square to rewrite in standard form.

Applications: Geometry of Quadrilaterals

• To determine if four points form a rectangle, check:

  • Opposite sides are equal and parallel.

  • All angles are right angles (use slopes or the Pythagorean Theorem).

Summary Table: Key Formulas and Properties