Podcast
Questions and Answers
What is the definition of an inequality?
What is the definition of an inequality?
A mathematical sentence that compares expressions.
Which of the following are inequality symbols?
Which of the following are inequality symbols?
- <
- >
- ≤
- ≥
- All of the above (correct)
What is the key phrase for ≤?
What is the key phrase for ≤?
- At least
- Less than or equal to (correct)
- Greater than
- More than
What is a solution of an inequality?
What is a solution of an inequality?
What is the set of all solutions called?
What is the set of all solutions called?
What does a closed circle represent on a graph?
What does a closed circle represent on a graph?
What does an open circle represent on a graph?
What does an open circle represent on a graph?
What is the subtraction property of inequalities?
What is the subtraction property of inequalities?
What is an example of the subtraction property?
What is an example of the subtraction property?
What does the addition property state?
What does the addition property state?
Provide an example of the addition property.
Provide an example of the addition property.
What does the multiplication and division property entail?
What does the multiplication and division property entail?
Provide an example for multiplication/division.
Provide an example for multiplication/division.
When multiplying both sides of an inequality by a negative number, do you flip the inequality symbol?
When multiplying both sides of an inequality by a negative number, do you flip the inequality symbol?
Dividing by a negative number follows the same rule as multiplying by a negative number.
Dividing by a negative number follows the same rule as multiplying by a negative number.
What is a multi-step inequality?
What is a multi-step inequality?
Provide an example of a multi-step inequality.
Provide an example of a multi-step inequality.
What should you do when you have variables on both sides of an inequality?
What should you do when you have variables on both sides of an inequality?
Provide an example with variables on both sides of an inequality.
Provide an example with variables on both sides of an inequality.
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Study Notes
Definition of Inequality
- An inequality is a mathematical sentence that expresses a relationship between two expressions.
Inequality Symbols
- Common symbols include:
- Less than: <
- Greater than: >
- Less than or equal to: ≤
- Greater than or equal to: ≥
Key Phrases for Inequality Symbols
- Greater than: indicates one quantity exceeds another.
- More than: synonymous with greater than.
- Less than or equal to (≤): implies a value is at most a certain limit.
- Greater than or equal to (≥): suggests a value is at least a certain threshold.
Solution of Inequality
- A solution of an inequality is a value that, when substituted into the inequality, results in a true statement.
Solution Set
- The collection of all possible solutions to an inequality is referred to as the solution set.
Closed Circle in Graphs
- A closed circle is used in graphing for inequalities with "greater than or equal to" (≥) and "less than or equal to" (≤) indicating that the endpoint value is included in the solution.
Open Circle in Graphs
- An open circle represents inequalities using "greater than" (>) or "less than" (<), indicating that the endpoint value is not included in the solution.
Subtraction Property of Inequalities
- When solving inequalities, subtracting the same number from both sides maintains the inequality.
Example of Subtraction Property
- For the inequality y + 8 ≤ 5:
- Subtracting 8 from both sides gives y ≤ -3.
Addition Property of Inequalities
- Adding the same number to both sides of an inequality does not change the inequality's direction.
Example of Addition Property
- For the inequality x - 6 ≥ -10:
- Adding 6 to both sides yields x ≥ -4.
Multiplication and Division Property
- Multiplying or dividing both sides of an inequality by the same non-zero number preserves the inequality's direction.
Example of Multiplication and Division
- For the inequality y ÷ 8 > 5:
- Multiplying both sides by 8 results in y > 40.
Multiplying by a Negative Number
- When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality symbol must be flipped to maintain a true statement.
Dividing by a Negative Number
- This follows the same rule as multiplying by a negative number: the inequality symbol flips direction.
Multi-step Inequality
- To solve multi-step inequalities:
- Simplify each side if necessary.
- Apply the order of operations in reverse.
Example of Multi-step Inequality
- For the inequality 2x - 4 ≥ 8:
- Adding 4 and dividing by 2 results in x ≥ 6.
Variables on Both Sides
- To solve inequalities with variables on both sides, rearranging is necessary to isolate the variables similarly to constants.
Example with Variables on Both Sides
- For the expression 6x - 5, isolate the variable as demonstrated in multi-step solutions.
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