Equations and Inequalities

• Equations and inequalities are mathematical statements used to represent relationships between numbers or variables
• Equations use an equals sign (=) to show that two expressions are equal
• Inequalities use inequality symbols (<, >, ≤, ≥) to show that two expressions are not equal

Adding and Subtracting Inequalities and Equations

• For equations, the same quantity can be added to or subtracted from both sides without changing the solution
• For inequalities, the same quantity can be added to or subtracted from both sides without changing the solution set
• Adding or subtracting the same value maintains the balance in equations and the direction of the inequality

Multiplying and Dividing Equations

• For equations, both sides can be multiplied or divided by the same non-zero quantity without changing the solution
• Multiplying or dividing by the same non-zero value maintains the balance in the equation

Multiplying and Dividing Inequalities

• When multiplying or dividing both sides of an inequality by a positive number, the direction of the inequality sign remains the same
• When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed to maintain the truth of the inequality

Graphing Inequalities on a Number Line

• Inequalities in one variable can be graphed on a number line
• Use an open circle (o) for strict inequalities (< or >) to indicate that the endpoint is not included
• Use a closed circle (•) for inclusive inequalities (≤ or ≥) to indicate that the endpoint is included
• Shade the region of the number line that represents all values satisfying the inequality

Graphing Inequalities on the Coordinate Plane

• Inequalities in two variables can be graphed on the coordinate plane
• Replace the inequality sign with an equals sign and graph the resulting equation as a boundary line
• Use a solid line for inclusive inequalities (≤ or ≥) to indicate that the boundary line is included in the solution
• Use a dashed line for strict inequalities (< or >) to indicate that the boundary line is not included in the solution
• Shade the region of the coordinate plane that represents all points satisfying the inequality
• Choose a test point not on the line to determine which side to shade. If the test point satisfies the inequality, shade the side containing the test point

Properties of Equations

• Addition Property of Equality: If a = b, then a + c = b + c
• Subtraction Property of Equality: If a = b, then a - c = b - c
• Multiplication Property of Equality: If a = b, then ac = bc
• Division Property of Equality: If a = b and c ≠ 0, then a/c = b/c
• Reflexive Property of Equality: a = a
• Symmetric Property of Equality: If a = b, then b = a
• Transitive Property of Equality: If a = b and b = c, then a = c
• Substitution Property of Equality: If a = b, then a can be substituted for b in any expression

Systems of Equations

• A system of equations is a set of two or more equations containing the same variables
• The solution to a system of equations is the set of values for the variables that satisfy all equations in the system simultaneously

Solving Systems of Equations by Graphing

• Graph each equation in the system on the coordinate plane
• The solution to the system is the point(s) of intersection of the graphs
• If the lines intersect at one point, the system has one solution
• If the lines are parallel and do not intersect, the system has no solution (inconsistent)
• If the lines are the same, the system has infinitely many solutions (dependent)

Solving Systems of Equations by Substitution

• Solve one equation for one variable in terms of the other variable(s)
• Substitute the expression for that variable into the other equation(s)
• Solve the resulting equation(s) for the remaining variable(s)
• Substitute the values found back into the original equation or the expression from step 1 to find the values of the other variable(s)

Solving Systems of Equations by Elimination (Addition/Subtraction)

• Multiply one or both equations by a constant so that the coefficients of one variable are opposites
• Add the equations together to eliminate one variable
• Solve the resulting equation for the remaining variable
• Substitute the value found back into one of the original equations to find the value of the other variable

Applications of Inequalities

• Inequalities are used to model and solve real-world problems involving constraints, limitations, or ranges of values
• Common applications include:
  • Budgeting and finance (e.g., spending limits, investment ranges)
  • Optimization problems (e.g., maximizing profit, minimizing cost)
  • Physics and engineering (e.g., tolerance limits, design constraints)
  • Statistics (e.g., confidence intervals)
  • Health and medicine (e.g., acceptable ranges for vital signs)

Steps for Solving Application Problems with Inequalities

• Read the problem carefully and identify the unknown quantity or variable
• Translate the problem into a mathematical inequality, using appropriate symbols and relationships
• Solve the inequality using algebraic techniques
• Interpret the solution in the context of the problem and state the answer with appropriate units
• Check if the solution makes sense in the real-world scenario