Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable appears only to the first power. Solving these equations involves isolating the variable.

• Definition: A linear equation has the form .

• Steps:

  1. Combine like terms.

  2. Isolate the variable by addition/subtraction.

  3. Divide by the coefficient of the variable.

• Example: Solve .

  • Add 7:

  • Divide by 3:

Solving Rational Equations

Rational equations contain fractions with variables in the denominator. Solutions require clearing denominators and checking for extraneous solutions.

• Definition: An equation involving rational expressions.

• Steps:

  1. Find a common denominator.

  2. Multiply both sides by the common denominator to clear fractions.

  3. Solve the resulting equation.

  4. Check for extraneous solutions.

• Example: Solve .

  • Multiply both sides by :

  • Divide by 4:

Solving Quadratic Equations

Quadratic equations are second-degree equations. They can be solved by factoring, taking square roots, or using the quadratic formula.

• Definition: An equation of the form .

• Methods:

  • Factoring: Express as a product of binomials and set each factor to zero.

  • Square Roots: Used when the equation is in the form .

  • Quadratic Formula:

• Example: Solve by factoring.

  • Factors:

  • Solutions: ,

Operations with Complex Numbers

Complex numbers are numbers of the form , where is the imaginary unit (). Operations include addition, subtraction, multiplication, and division.

• Definition: , where and are real numbers.

• Key Operations:

  • Add/Subtract: Combine real and imaginary parts separately.

  • Multiply: Use distributive property and .

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

• Example:

Solving Radical Equations

Radical equations contain variables under a root. Solutions require isolating the radical and raising both sides to the appropriate power.

• Definition: An equation with a variable inside a radical.

• Steps:

  1. Isolate the radical.

  2. Raise both sides to the power that eliminates the radical.

  3. Solve the resulting equation.

  4. Check for extraneous solutions.

• Example: Solve .

  • Square both sides:

Solving Polynomial Equations

Polynomial equations involve terms with variables raised to whole number powers. Some equations are quadratic in form and can be solved by substitution.

• Definition: An equation involving a polynomial expression.

• Methods:

  • Factoring

  • Substitution (for quadratic form)

  • Using the Rational Root Theorem (advanced)

• Example: Solve by substitution.

  • Let , so

  • Factor:

  • or ; or

  • ,

Solving Linear Inequalities

Linear inequalities are similar to linear equations but involve inequality signs. Solutions are often expressed in set notation, interval notation, and as a graph.

• Definition: An inequality of the form (or , , ).

• Steps:

  1. Solve as a linear equation.

  2. Express the solution in set and interval notation.

  3. Graph the solution on a number line.

• Example: Solve .

  • Add 3:

  • Divide by 2:

  • Interval notation:

Solving Combined Inequalities

Combined inequalities involve two inequalities joined by 'and' or 'or'. Solutions are often intervals.

• Example: Solve .

  • Solution: is between 1 and 4.

  • Interval notation:

Solving Absolute Value Equations and Inequalities

Absolute value equations and inequalities require considering both positive and negative cases.

• Definition: means or .

• Example: Solve .

  • or

  • or

• Absolute Value Inequality: means .

Graphs and Analytic Geometry

Distance Between Two Points

The distance formula calculates the length between two points in the coordinate plane.

• Formula:

• Example: Find the distance between and .

Midpoint Between Two Points

The midpoint formula finds the point exactly halfway between two points.

• Formula:

• Example: Find the midpoint between and .

Intercepts and Symmetry of Graphs

Intercepts are points where a graph crosses the axes. Symmetry can be with respect to the x-axis, y-axis, or origin.

• x-intercept: Where the graph crosses the x-axis ().

• y-intercept: Where the graph crosses the y-axis ().

• Symmetry:

  • x-axis: Replace with ; if unchanged, symmetric about x-axis.

  • y-axis: Replace with ; if unchanged, symmetric about y-axis.

  • Origin: Replace with and with ; if unchanged, symmetric about origin.

• Example: The graph of is symmetric about the y-axis.

Linear Functions and Equations of Lines

Slope of a Line

The slope measures the steepness of a line.

• Formula:

• Example: Find the slope between and .

Equations of a Line

There are several forms for the equation of a line, depending on the information given.

• Slope-Intercept Form:

• Point-Slope Form:

• General Form:

• Standard Form:

• Example: Find the equation of a line with slope $2(1, 3)$.

  • Point-slope:

  • Simplify:

Finding Intercepts and Sketching Graphs

To sketch a line, find its intercepts and plot them.

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

• Example: For , y-intercept is $1x = -0.5$.

Equations of Circles

Standard and General Forms

The equation of a circle can be written in standard or general form.

• Standard Form:

• General Form:

• Example: Center , radius $3$:

  • Standard:

  • Expand to general form:

Summary Table: Key Formulas