Glossary
"Undo" Operations
The process of using inverse mathematical operations (e.g., subtraction for addition, division for multiplication) to isolate a variable in an equation or inequality.
Example:
To solve , you would use the "undo" operation of adding 8 to both sides.
Constraints
Limitations or restrictions given in a real-world problem that must be considered when forming or interpreting mathematical models and their solutions.
Example:
If a problem states that a budget is limited to 50 is a constraint on your spending.
Distributive Property
An algebraic property stating that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products, expressed as $$a(b+c) = ab + ac$$.
Example:
When simplifying , you apply the distributive property to get .
Feasibility
The quality of a solution being practical and making sense within the context of a real-world problem.
Example:
A solution of "-3 apples" would lack feasibility in a problem about fruit quantities.
Flipping the Inequality Sign
The crucial rule in solving inequalities that requires reversing the direction of the inequality symbol when multiplying or dividing both sides by a negative number.
Example:
When solving , you must divide by -2 and flip the inequality sign to get .
Infinitely Many Solutions (Identity)
A type of linear equation where any real value of the variable will satisfy the equation, leading to a true statement (e.g., $$0 = 0$$) after simplification.
Example:
The equation simplifies to , meaning it has infinitely many solutions.
Interval Notation
A concise way to represent a set of real numbers, especially solution sets of inequalities, using parentheses for strict inequalities and brackets for inclusive ones.
Example:
The inequality is represented as in interval notation.
Linear Equation
An algebraic equation where the highest power of the variable is one, resulting in a straight line when graphed. Its standard form is $$ax + b = c$$.
Example:
Solving for x is an example of working with a linear equation.
Linear Inequality
A mathematical statement that compares two expressions using an inequality symbol ($$<, >, \leq, \geq$$), representing a range of possible solutions.
Example:
The statement is a linear inequality.
Modeling (Real-World Applications)
The process of translating a real-life situation or problem into a mathematical equation or inequality to find a solution.
Example:
Writing the equation to represent the cost of a taxi ride based on miles is an example of modeling.
No Solution (Inconsistent)
A type of linear equation where no value of the variable can satisfy the equation, leading to a false statement (e.g., $$0 = 5$$) after simplification.
Example:
The equation simplifies to , indicating there is no solution.
Number Line (representation)
A visual representation of the solution set of an inequality, using open or closed circles and shading to indicate the range of values.
Example:
To show , you would draw an open circle at 4 and shade to the left on a number line.
One Solution
A type of linear equation where there is a single, unique value for the variable that satisfies the equation.
Example:
The equation has one solution, which is .
Set-Builder Notation
A mathematical notation used to describe a set by specifying the properties that its members must satisfy.
Example:
The set of all numbers greater than 3 can be written as using set-builder notation.
Solution (of an equation)
The specific value(s) of the variable that make a given equation true. For linear equations, there is typically one unique solution.
Example:
In the equation , the solution is .
Solution Set
The collection of all values for the variable that make an inequality true. Unlike equations, this is often a range of numbers.
Example:
For the inequality , the solution set includes all numbers greater than 5.
Units
Standard measures (e.g., dollars, hours, miles) that must be included with numerical answers in real-world problems to provide context and meaning.
Example:
If a problem asks for time, the answer should include units like "5 hours" or "30 minutes."
Verification
The process of checking if a calculated solution is correct by substituting it back into the original equation or inequality to see if it yields a true statement.
Example:
After solving to get , you perform verification by plugging 5 back in: , which is true.