Grade 8 Mathematics Inequality PDF

Republic of the Philippines Department of Education National Capital Region Schools Division Office...

Republic of the Philippines Department of Education National Capital Region Schools Division Office Nagpayong High School SUPREME SECONDARY LEARNER GOVERNMENT Centennial II, Nagpayong Brgy. Pinagbuhatan Pasig City GRADE 8 Mathematics Contributors: Mary Anthoneth N. Raco Merry Angeline Mendez FOR THE YOUTH, BY THE YOUTH INEQUALITY 2. Substitute known values into the Inequality is a mathematical statement inequality. 3. Solve for the unknown variable. where one expression is not equal to 4. State the answer in the context of the another. problem. It uses the symbols, ≤, ≥, or #. The graph of a linear inequality in two EQUATION uses the symbol = variables shows all points (x, y) that indicating that the value of the satisfy the inequality. A line divides the expressions from both sides were equal. coordinate plane into two half-planes. To Differences between linear equation and graph a linear inequality: linear inequalities in two variables in 1. Replace the inequality symbol with an terms of: equals sign to find the boundary line. 1. definition 2. Graph the boundary line. Use a solid 2. standard form line for ≤ or ≥, and a dashed line for < 3. symbol used or >. 4. translating verbal statements to 3. Test points in each half-plane to mathematical statements determine which satisfies the inequality. 5. characteristic of the graphs 4. Shade the half-plane containing the 6. effects when multiplied/divided by a points that satisfy the inequality. negative integer System of linear inequalities - a group A linear inequalities can be written in of linear inequalities how to solve a one of the following forms: system of linear inequalities? Ax+ByC inequalities, graph each inequality on Ax+By_C is the overlapping shaded region, Where A, B and C are real numbers and representing all points that satisfy all A and B are both not equal to zero. inequalities simultaneously. Solutions of linear inequalities can also Relation - A relation is a set of ordered be expressed in ordered pairs. pairs (x, y), where x is the input An ordered pair (x,y) is also a solution of (domain) and y is the output (range). of an inequalities x and y if a true The domain is the set of all x-values, statement results when variable in the and the range is the set of all y-values. equality is replaced by the coordinate of Relations can be classified into four ordered pairs. types based on the mapping between To solve problems with linear the domain and range: inequalities: One-to-one: Each element in the 1. Translate the problem into a domain corresponds to exactly one mathematical inequality. Address : Centennial II-B, Nagpayong, Pinagbuhatan, Pasig City Phone : 8573-1475 Doc. Code. Title SSLG- Rev 00 REVIEWER- Email : [email protected] Q2-G8-MATH Facebook : NHS Supreme Secondary Learner Government Status Page 1 of 3  Republic of the Philippines Department of Education National Capital Region Schools Division Office Nagpayong High School SUPREME SECONDARY LEARNER GOVERNMENT Centennial II, Nagpayong Brgy. Pinagbuhatan Pasig City GRADE 8 Mathematics Contributors: Mary Anthoneth N. Raco Merry Angeline Mendez FOR THE YOUTH, BY THE YOUTH element in the range (e.g., one shoe per Dependent variable: Its value is foot). determined by one or more other One-to-many: One element in the variables. It's the outcome that depends domain can correspond to multiple on other factors. elements in the range (e.g., one student Independent variable: Its value is not can take many subjects). affected by other variables in the Many-to-one: Multiple elements in the equation or function. It's the variable that domain correspond to a single element influences the dependent variable. in the range (e.g., many students can A function's domain is the set of all have the same teacher). possible input values (x-values), and its Many-to-many: Multiple elements in the range is the set of all possible output domain correspond to multiple elements values (y-values). These can be in the range (e.g., many parents can identified from the function's ordered have many children). pairs, mapping diagram, table, graph, or Representation - relation may be rule. represented using table of valves, Linear function - A linear function is a mapping dagram, graph and equation or function of the form f(x) = mx + b, where rule. m and b are real numbers, and m and Table- A table of values lists pairs of x f(x) are not both zero. In this equation, m and y values that satisfy a given represents the slope, and b represents equation, illustrating the relationship the y-intercept. between x and y. Linear functions are useful for Mapping diagram - A mapping diagram modeling various real-world situations, uses two columns (domain and range) such as calculating telephone charges, to visually represent a relation, population growth, event costs, rentals, connecting paired elements with lines. and rates of change. Successfully The diagram can be rectangular or applying linear functions to word circular. problems involves these steps: Graph - A graph visually represents the 1. Identify given information. relationship between x and y values 2. Determine the dependent and using points and lines. independent variables. Rule - An equation or rule describes 3. Create a table of values. how one variable (like y) depends on 4. Find the slope (m) and y-intercept (b), another (like x). For example, y = 2x then write the linear equation (y = mx + means y is always twice. b). Vertical line test - A graph represents a 5. Graph the linear equation. function ip and only if a vertical line 6. Answer the question posed in the word intersects the graph at exactly one point. problem. There are two types of variables. Address : Centennial II-B, Nagpayong, Pinagbuhatan, Pasig City Phone : 8573-1475 Doc. Code. Title SSLG- Rev 00 REVIEWER- Email : [email protected] Q2-G8-MATH Facebook : NHS Supreme Secondary Learner Government Status Page 2 of 3  Republic of the Philippines Department of Education National Capital Region Schools Division Office Nagpayong High School SUPREME SECONDARY LEARNER GOVERNMENT Centennial II, Nagpayong Brgy. Pinagbuhatan Pasig City GRADE 8 Mathematics Contributors: Mary Anthoneth N. Raco Merry Angeline Mendez FOR THE YOUTH, BY THE YOUTH Many conditional statements, even if not syllogism, where two premises sharing a initially written in "if-then" form, can be common element lead to a conclusion rewritten using this structure. The (e.g., If A=B and B=C, then A=C). subject of the sentence becomes the Writing proofs - A proof is a logical hypothesis (the "if" part), and the argument where each statement is predicate becomes the conclusion (the supported by given information, "then" part). definitions, axioms, postulates, Inverse - The inverse of a conditional theorems, or previously proven statement ("if p, then q") is formed by statements. negating both the hypothesis (p) and the A postulate is a statement accepted as conclusion (q), resulting in "if not p, then true without proof, while a theorem is a not q" (or symbolically, ~p → ~q). statement proven deductively. There are Negating a statement means stating its two main types of proof: direct and opposite. indirect. A direct proof is a series of Contrapostive - The contrapositive of a statements, each justified by previous conditional statement ("if p, then q") is statements or given information, leading created by switching the hypothesis and to the conclusion. conclusion, and then negating both. This Paragraph form - One way to write a results in "if not q, then not p" (or proof is using paragraph form, where symbolically, ~q → ~p). you explain in a coherent paragraph why Converse - The converse of a a conjecture about a given situation is conditional statement ("if p, then q") is true. formed by simply switching the Two column proof - A two-column hypothesis (p) and the conclusion (q), proof (or formal proof) presents a proof resulting in "if q, then p" (or symbolically, in two columns: one for statements and q → p). one for the justifications of those Inductive and deductive reasoning statements. are essential methods of reasoning in Indirect proof - An indirect proof (often mathematics. They form the basis of written in paragraph form) works by theorems and proofs, and continue to be assum used in mathematical discovery. Inductive Reasoning This kind of reasoning uses specific examples to arrive at a general rule, generalizations, or conclusions. Deductive Reasoning - Deductive reasoning uses established facts, properties, and laws to logically reach a conclusion. A common example is the Address : Centennial II-B, Nagpayong, Pinagbuhatan, Pasig City Phone : 8573-1475 Doc. Code. Title SSLG- Rev 00 REVIEWER- Email : [email protected] Q2-G8-MATH Facebook : NHS Supreme Secondary Learner Government Status Page 3 of 3