Inequality Fundamentals

Questions and Answers

What is the definition of an inequality?

A mathematical sentence that compares expressions.

Which of the following are inequality symbols?

≤

≥

All of the above (correct)

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?

A value that makes the inequality true. Signup and view all the answers

What is the set of all solutions called?

Solution set. Signup and view all the answers

What does a closed circle represent on a graph?

It represents when using greater than or equal to and less than or equal to. Signup and view all the answers

What does an open circle represent on a graph?

It represents greater than or just less than. Signup and view all the answers

What is the subtraction property of inequalities?

Subtract the same number from both sides. Signup and view all the answers

What is an example of the subtraction property?

y + 8 ≤ 5 leads to y ≤ -3 after subtracting 8. Signup and view all the answers

What does the addition property state?

Adding the same number to each side of the inequality. Signup and view all the answers

Provide an example of the addition property.

x - 6 ≥ -10 leads to x ≥ -4 after adding 6. Signup and view all the answers

What does the multiplication and division property entail?

Multiply or divide both sides by the same number. Signup and view all the answers

Provide an example for multiplication/division.

y ÷ 8 > 5 leads to y > 40 after multiplying both sides by 8. Signup and view all the answers

When multiplying both sides of an inequality by a negative number, do you flip the inequality symbol?

True Signup and view all the answers

Dividing by a negative number follows the same rule as multiplying by a negative number.

True Signup and view all the answers

What is a multi-step inequality?

Simplify each side if necessary and use order of operations in reverse. Signup and view all the answers

Provide an example of a multi-step inequality.

2x - 4 ≥ 8 leads to x ≥ 6 after simplifying. Signup and view all the answers

What should you do when you have variables on both sides of an inequality?

Get variables on their own sides, same as for the constants. Signup and view all the answers

Provide an example with variables on both sides of an inequality.

6x - 5 leads to isolating x. Signup and view all the answers

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.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Explore the concept of inequalities, including their symbols and interpretations. This quiz covers key phrases, solution sets, and graphing techniques related to inequalities. Test your understanding and improve your mathematical skills regarding inequalities.