Solving Equations and Inequalities

Solving Equations

• Equations state equality between two expressions.
• To solve an equation, isolate the variable using inverse operations (addition, subtraction, multiplication, division).
• Always perform the same operation on both sides of the equation to maintain equality.
• Check your solution by substituting it back into the original equation.
• Example: Solve for x: 2x + 5 = 11
  • Subtract 5 from both sides: 2x = 6
  • Divide both sides by 2: x = 3

Solving Inequalities

• Inequalities compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), ≠ (not equal to).
• Solving inequalities is similar to solving equations, but the direction of the inequality sign changes when multiplying or dividing by a negative number.
• Graphing inequalities on a number line involves shading the region representing the solution.
• Example: Solve for x: 3x - 7 ≥ 8
  • Add 7 to both sides: 3x ≥ 15
  • Divide both sides by 3: x ≥ 5

Literal Equations

• Literal equations involve more than one variable.
• Solving for a particular variable in a literal equation means isolating that variable on one side of the equation.
• Example: Solve for y in the equation 2x + 3y = 9
  • Subtract 2x from both sides: 3y = 9 - 2x
  • Divide both sides by 3: y = (9 - 2x) / 3

Domain, Range, and Functions

• A function is a relationship where each input has exactly one output.
• The domain of a function is the set of all possible input values (x-values).
• The range of a function is the set of all possible output values (y-values).
• Functions can be represented by tables, graphs, or equations.
• Identifying domain and range from a graph involves examining the x-values and y-values the graph occupies.

Writing and Evaluating Functions

• Writing functions involves expressing a relationship in mathematical form (e.g., f(x) = 2x + 1).
• Evaluating a function involves substituting a value for the input variable (x) and calculating the output (f(x)).
• Example: Given the function f(x) = x² - 3x + 2, evaluate f(4).
  • Substitute x = 4: f(4) = (4)² - 3(4) + 2
  • Calculate: f(4) = 16 - 12 + 2 = 6

Graphing & Equation Forms

• Graphing linear equations involves plotting points that satisfy the equation and connecting them to form a straight line.
• Different forms of linear equations include slope-intercept form (y = mx + b), point-slope form, and standard form.
• Slope-intercept form makes it easy to identify the slope (m) and y-intercept (b).
• Graphing linear inequalities involves shading the region representing the inequality.
• Point-slope equation: y - y₁ = m(x - x₁) - Used with a given point and the slope of a line.
• Standard form of a linear equation: Ax + By = C - Useful for finding intercepts and determining if the line describes a proportional relationship.

Systems of Equations

• A system of equations consists of two or more equations with the same variables.
• Solving a system of equations means finding the values of the variables that satisfy all the equations in the system.
• Methods for solving systems of equations include:
  • Substitution: Solve one equation for a variable and substitute the expression into the other equation.
  • Elimination: Add or subtract equations to eliminate one variable. This results in an equation with one variable which can be solved.
  • Graphing: Graph each equation and find the intersection point(s), if any.
• Example: Solve the system of equations by substitution: x + y = 5 2x - y = 4
  • Solve the first equation for y: y = 5 - x
  • Substitute into second equation: 2x - (5 - x) = 4
  • Simplify and solve for x: 3x = 9, x = 3
  • Substitute x = 3 back into the first equation to solve for y: 3 + y = 5, y = 2
• Systems of equations can have one solution, infinitely many solutions (same line), or no solutions (parallel lines).

This quiz covers the fundamental concepts of solving equations and inequalities. You'll learn how to isolate variables, maintain equality, and understand the effects of operations on inequalities. Practice your skills with examples and enhance your problem-solving ability.