Glossary
At least (≥)
A key phrase indicating that a value must be greater than or equal to a specified number.
Example:
You need to score at least 70% on the exam to pass, meaning your score (S) must satisfy S ≥ 70.
At most (≤)
A key phrase indicating that a value must be less than or equal to a specified number.
Example:
The elevator has a weight limit of at most 1000 pounds, so the total weight (W) must be W ≤ 1000.
Closed Circle
A filled-in circle (●) used on a number line graph to indicate that an endpoint is included in the solution set (used for ≤ or ≥).
Example:
To graph x ≤ 10, a closed circle is drawn at 10, signifying that 10 is a valid solution.
Constraint
A limitation or restriction that defines the possible values for variables in a problem, often expressed as an inequality.
Example:
A baker has only 500 grams of flour, which is a constraint on the total amount of dough they can make.
Flip the Inequality Sign
The crucial rule that requires reversing the direction of the inequality symbol when multiplying or dividing both sides of an inequality by a negative number.
Example:
When solving -3x < 12, you must flip the inequality sign to x > -4 after dividing by -3.
Isolate the Variable
The algebraic process of manipulating an inequality to get the variable by itself on one side of the inequality symbol.
Example:
To solve 2x + 5 > 15, you first subtract 5 from both sides to begin to isolate the variable x.
Less than (<)
A key phrase indicating that a value must be strictly less than a specified number.
Example:
The temperature must be less than 0°C for the water to freeze, so T < 0.
Linear Inequality
A mathematical statement that compares two expressions using an inequality symbol (>, <, ≥, ≤), where the highest power of the variable is one.
Example:
To qualify for a scholarship, a student's GPA must be at least 3.5, which can be written as GPA ≥ 3.5.
More than (>)
A key phrase indicating that a value must be strictly greater than a specified number.
Example:
To win the game, you need more than 50 points, meaning your score (S) must be S > 50.
Number Line
A visual representation used to graph the solution set of an inequality, showing all real numbers that satisfy the condition.
Example:
Graphing x ≤ 7 on a number line involves placing a closed circle at 7 and shading to the left.
Open Circle
A hollow circle (o) used on a number line graph to indicate that an endpoint is not included in the solution set (used for < or >).
Example:
When graphing x > 3, an open circle is placed at 3 to show that 3 is not part of the solution.
Real-World Interpretation
The process of translating a mathematical solution back into the context of the original word problem, ensuring the answer makes practical sense.
Example:
If solving for 'number of people' yields x > 4.7, the real-world interpretation would be 'at least 5 people' since you can't have a fraction of a person.
Shading (on Number Line)
The process of highlighting a section of the number line to represent all the values that satisfy an inequality.
Example:
For x < 5, you would use an open circle at 5 and shading to the left to show all numbers less than 5.
Solution Interval
The range of values that satisfy a given inequality, often expressed using interval notation or by graphing on a number line.
Example:
For the inequality x > 5, the solution interval includes all numbers greater than 5, but not 5 itself.
Variable
A symbol, typically a letter, that represents an unknown quantity or a value that can change within a problem.
Example:
In the problem 'A budget of $500 for x items,' x is the variable representing the number of items.