Grade 8 Mathematics - Inequality and Variables
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Questions and Answers

What defines the overlap in a system of linear inequalities?

  • The area with no inequality solutions
  • The overlapping shaded region representing all solutions (correct)
  • The region outside all inequalities
  • The union of all shaded regions

Which of the following statements is true about ordered pairs in linear inequalities?

  • An ordered pair is a solution if it satisfies the inequality. (correct)
  • An ordered pair can only be a solution if the statement is an equation.
  • An ordered pair cannot be a solution if both values are positive.
  • An ordered pair must contain integers to be a solution.

What should you do first to solve a problem involving linear inequalities?

  • Assume values for x and y.
  • Graph the inequalities directly without calculations.
  • Translate the problem into a mathematical inequality. (correct)
  • Eliminate one of the variables algebraically.

How can relations be classified based on their mappings?

<p>By the relationship between x-values and y-values. (B)</p> Signup and view all the answers

What is the domain in a relation?

<p>The set of all x-values. (C)</p> Signup and view all the answers

What is described as the value that is determined by one or more other variables?

<p>Dependent variable (A)</p> Signup and view all the answers

Which term describes a relationship where one element in the domain corresponds to multiple elements in the range?

<p>One-to-many (A)</p> Signup and view all the answers

In a function, what is the set of all possible input values called?

<p>Domain (C)</p> Signup and view all the answers

What describes the variable that influences the dependent variable in an equation?

<p>Independent variable (A)</p> Signup and view all the answers

Which relationship is characterized by multiple elements in the domain corresponding to multiple elements in the range?

<p>Many-to-many (A)</p> Signup and view all the answers

What is the term for when one student can take many subjects?

<p>One-to-many (B)</p> Signup and view all the answers

In a functional relationship, the outcome is based on what kind of variable?

<p>Dependent variable (D)</p> Signup and view all the answers

What type of relationship has multiple elements in the domain corresponding to a single element in the range?

<p>Many-to-one (A)</p> Signup and view all the answers

Which symbol is NOT used in mathematical inequalities?

<p>= (B)</p> Signup and view all the answers

What type of line is used to graph the inequality $y < 2x + 3$?

<p>Dashed line (C)</p> Signup and view all the answers

In the process of graphing a linear inequality, which step follows graphing the boundary line?

<p>Test points to determine which half-plane to shade (D)</p> Signup and view all the answers

Which of the following correctly represents a linear inequality?

<p>3y - 2 &lt; 5 (D)</p> Signup and view all the answers

What is the purpose of using a solid line when graphing inequalities?

<p>It shows the range of solutions includes the boundary (A)</p> Signup and view all the answers

What must be done first when solving a linear inequality?

<p>State the problem (C)</p> Signup and view all the answers

Which characteristic is different between linear equations and linear inequalities?

<p>Use of equality and inequality symbols (D)</p> Signup and view all the answers

When graphing the linear inequality $y ≥ 3x - 2$, what should be done with the boundary line?

<p>It should be a solid line (A)</p> Signup and view all the answers

What is a linear function represented by?

<p>f(x) = mx + b (A)</p> Signup and view all the answers

What does the 'm' in the linear equation f(x) = mx + b represent?

<p>The slope (A)</p> Signup and view all the answers

Which of the following methods can represent a relation?

<p>Mapping diagram (A)</p> Signup and view all the answers

What is the first step in applying linear functions to word problems?

<p>Identify given information (B)</p> Signup and view all the answers

What does the vertical line test determine about a graph?

<p>If it's a linear function (C)</p> Signup and view all the answers

Which of the following accurately describes the y-intercept in a linear function?

<p>The value of y when x=0 (D)</p> Signup and view all the answers

What is the purpose of creating a table of values in a linear function?

<p>To list pairs of x and y values (D)</p> Signup and view all the answers

Which of the following is NOT a method for representing a relation?

<p>Schematic diagram (B)</p> Signup and view all the answers

What is a conditional statement typically structured as?

<p>An 'if-then' statement (D)</p> Signup and view all the answers

How can two premises A and B lead to a conclusion C?

<p>If A=B and B=C, then A=C (D)</p> Signup and view all the answers

What does the inverse of a conditional statement involve?

<p>Negating both the hypothesis and conclusion (A)</p> Signup and view all the answers

Which statement is accepted as true without proof?

<p>An axiom (A)</p> Signup and view all the answers

What is a theorem?

<p>A statement proven deductively (B)</p> Signup and view all the answers

What part of a conditional statement is represented by the hypothesis?

<p>The 'if' part (C)</p> Signup and view all the answers

Which of the following best describes a proof?

<p>A logical argument supported by given information (B)</p> Signup and view all the answers

What is the formal structure of the inverse of 'if p, then q'?

<p>not p implies not q (D)</p> Signup and view all the answers

What is the contrapositive of the statement 'if p, then q'?

<p>if not q, then not p (A)</p> Signup and view all the answers

Which of the following best describes inductive reasoning?

<p>Drawing conclusions from specific instances (B)</p> Signup and view all the answers

Which proof format involves writing statements and justifications in two separate columns?

<p>Two-column proof (C)</p> Signup and view all the answers

What is the definition of a direct proof?

<p>A proof consisting of statements justified by previous information (D)</p> Signup and view all the answers

What is the converse of the statement 'if p, then q'?

<p>if q, then p (B)</p> Signup and view all the answers

What does deductive reasoning primarily rely on?

<p>Established facts and laws (B)</p> Signup and view all the answers

In indirect proof, what is the general approach taken?

<p>Assuming the negation of the conclusion and proving it leads to a contradiction (C)</p> Signup and view all the answers

Which reasoning method uses specific examples to reach a general conclusion?

<p>Inductive reasoning (D)</p> Signup and view all the answers

Flashcards

Inequality

A mathematical statement where one expression is not equal to another. It uses symbols like ≤, ≥, or ≠.

Equation

A mathematical statement showing equality between two expressions, using the symbol =.

Linear inequality in two variables

An inequality that can be written in the form Ax + By ≤ c (or ≥, <, or >).

Graph of a linear inequality

Shows all points (x, y) that satisfy the inequality on a coordinate plane.

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Boundary line

The line that separates the half-planes when graphing a linear inequality.

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Solid line

Used to graph inequalities with ≤ or ≥, indicating the points on the line are part of the solution.

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Dashed line

Used to graph inequalities with < or >, indicating the points on the line are not part of the solution.

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Solving a linear inequality

Finding the value(s) of the unknown variable(s) that satisfy the inequality.

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Linear Inequality

An inequality that can be written in the form Ax + By ≤ C, Ax + By ≥ C, Ax + By < C, or Ax + By > C, where A, B, and C are real numbers and A and B are not both zero.

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System of Linear Inequalities

A group of two or more linear inequalities solved simultaneously.

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Solution of a Linear Inequality

An ordered pair (x, y) that makes the inequality true when substituting the values.

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Solving System of Linear Inequalities

Finding the region in a graph where the shaded areas of each inequality overlap; this region contains all the solutions.

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Function

A special kind of relation where each input has exactly one output.

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Domain

The set of all possible input values (x-values) for a function.

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Range

The set of all possible output values (y-values) for a function.

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One-to-one function

Each input has exactly one unique output, and each output corresponds to only one input.

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Many-to-one function

Multiple inputs can have the same output.

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Independent variable

The input or cause in a function. Its value is not influenced by other variables.

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Dependent variable

The output or effect in a function. Its value is determined by the independent variable.

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Relation Representation

A relation can be shown using different methods like tables, mapping diagrams, graphs, and equations. Each method highlights different aspects of the relationship between variables.

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Table of Values

A table lists pairs of x and y values that satisfy a given equation, showcasing the relationship between them.

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Mapping Diagram

A visual representation using two columns (domain and range) to connect paired elements with lines, showing how input values relate to output values.

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Graph

A visual representation showing the relationship between x and y values using points and lines on a coordinate plane.

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Linear Function

A function with a straight-line graph. It's represented as f(x) = mx + b, where m is the slope and b is the y-intercept.

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Slope

Describes how steep a line is. It's the change in y divided by the change in x (rise over run).

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Y-intercept

The point where the line crosses the vertical y-axis. It's the value of y when x is zero.

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Vertical Line Test

A test to determine if a graph represents a function. If a vertical line intersects the graph at only one point, it's a function.

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Negating a statement

Expressing the opposite of a statement. For example, the negation of "the sun is shining" is "the sun is not shining."

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Direct proof

A series of logical steps, starting from known facts and leading to the desired conclusion.

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Contrapositive

Switching the hypothesis and conclusion of a conditional statement and negating both parts. For example, the contrapositive of "if it is raining, then the ground is wet" is "if the ground is not wet, then it is not raining."

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Converse

Switching the hypothesis and conclusion of a conditional statement. For example, the converse of "if it is raining, then the ground is wet" is "if the ground is wet, then it is raining."

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Inductive reasoning

Using specific examples to come up with a general rule or conclusion.

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Deductive reasoning

Using established facts and logic to arrive at a conclusion.

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Indirect proof

A proof that starts by assuming the opposite of what you want to prove, then showing that this leads to a contradiction. This means your original assumption must be false, proving the original statement.

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Two-column proof

A formal proof typically used for geometric proofs, with one column for statements and another for justifications.

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Conditional Statement

A statement that makes a claim about the relationship between two parts: a hypothesis and a conclusion, usually in the form 'If p, then q'.

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Hypothesis

The 'if' part of a conditional statement, representing the condition that must be true for the conclusion to follow.

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Conclusion

The 'then' part of a conditional statement, representing the outcome that follows if the hypothesis is true.

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Syllogism

A logical argument with two premises sharing a common element, leading to a conclusion. For example, 'If A=B and B=C, then A=C'.

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Proof

A logical argument where each statement is supported by given information, definitions, axioms, postulates, theorems, or previously proven statements.

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Postulate

A statement accepted as true without proof. It's a fundamental building block of a system.

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Theorem

A statement proven deductively using definitions, axioms, postulates, or previously proven statements.

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Study Notes

Grade 8 Mathematics - Inequality

  • Inequality is a statement where one expression is not equal to another
  • Inequality symbols include ≤, ≥, ≠
  • Equation uses the = symbol
  • Linear inequalities can be written as Ax + By < C, Ax + By > C, Ax + By ≤ C, Ax + By ≥ C, where A, B, and C are real numbers and A and B are not both zero.
  • In graphing a linear inequality: replace the inequality symbol with an equals sign to find the boundary line, graph the boundary line (solid line for ≤ or ≥, dashed line for < or >), and test points in each half-plane to determine which satisfies the inequality. Shade the half-plane containing the points that satisfy the inequality.
  • A system of linear inequalities is a group of linear inequalities solved simultaneously.

Grade 8 Mathematics - Variables

  • Variables can be independent or dependent
  • A dependent variable's value is determined by other variables.
  • An independent variable's value is not affected by other variables.
  • A function's domain is the set of possible input values (x-values), and its range is the set of possible output values (y-values).
  • Linear functions are of the form f(x) = mx + b, where m is the slope and b is the y-intercept.

Grade 8 Mathematics - Representation

  • Relations can be represented using tables, mapping diagrams, and graphs.
  • A table of values lists pairs of x and y values satisfying an equation.
  • Mapping diagrams visually represent relations connecting paired elements.
  • A graph visually represents the relationship between x and y values using points and lines.

Grade 8 Mathematics - Conditional Statements

  • Conditional statements can be expressed as "if-then" statements.
  • The subject of the sentence is the hypothesis (the "if" part), and the predicate is the conclusion (the "then" part).
  • The inverse of a conditional statement negates both the hypothesis and conclusion.
  • The contrapositive switches and negates both the hypothesis and conclusion.
  • The converse switches the hypothesis and conclusion.

Grade 8 Mathematics - Reasoning

  • Inductive reasoning uses specific examples to reach a general rule.
  • Deductive reasoning uses facts, properties, and laws to reach a conclusion.
  • A syllogism is a form of deductive reasoning where two premises lead to a conclusion.
  • Proofs are logical arguments supporting a statement. Direct proofs are a series of statements justified by previous statements. Indirect proofs work by assuming the opposite to arrive at a contradiction.

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Grade 8 Mathematics Inequality PDF

Description

This quiz covers key concepts in Grade 8 Mathematics related to inequalities and variables. It includes definitions, symbols, and graphing techniques for linear inequalities, as well as the distinctions between independent and dependent variables. Test your understanding of these important mathematical principles.

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