MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

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1 FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) INEQUALITY = a mathematical statement that contains one of these four inequalit signs: <,, >,. These inequalit signs indicate that two epressions have different values. REVIEW OF INEQUALITIES I) AN INEQUALITY IS A MATHEMATICAL STATEMENT CONTAINING ONE OF THESE FOUR SIGNS: <,, >, LOCATED BETWEEN TWO DIFFERENT EXPRESSIONS. Inequalities alwas contain one of these four signs: <,, >,, which indicates that the epression on one side of the inequalit sign is a different value, either greater (larger) than or less (smaller) than, the epression on the other side. e.g. INEQUALITY < 3 MEANING (LS = Left Side; RS = Right Side) LS < RS; or the LS epression is less (smaller) than the RS epression. > + 3 LS > RS; or the LS epression is greater (bigger) than the RS epression. LS RS; or the LS epression is equal to or less (smaller) than the RS epression. LS RS; or the LS epression is equal to or greater (bigger) than the RS epression. II) INEQUALITIES IN ONE VARIABLE A) Inequalities in one variable have onl one kind of variable. The eamples below are inequalities in one variable. Stud them carefull: NOTICE that the all have onl one kind of variable. Be sure ou understand and memorize how to recognize them: e.g. > 3 ; ; > ; w 9. ; a < 3. ; B) Since an inequalit has one side that is a greater or smaller value than the other side, it has man possible solutions that are best illustrated with a graph. Since inequalities in one variable have onl one tpe of variable, the are graphed on one AXIS, the number line. GRAPHING INEQUALITIES I) Inequalities are solved b the same process that is used to solve equations with one additional rule: WHEN AN INEQUALITY IS MULTIPLIED OR DIVIDED BY A NEGATIVE NUMBER THE INEQUALITY SIGN ( <,, >, ) IS FLIPPED AROUND: i.e. < becomes > ; becomes. A) SAMPLE PROBLEMS 1: Stud these eamples carefull. Be sure ou understand and memorize the process used to complete them. 1) Solve ( ) ) Solve 3 >19 3) Solve < >1 ( ) 3 3 > 1 < 1< ( 1) < ( ) > < R. Ashb 017. Duplication b permission onl.

2 FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION - 1 B) REQUIRED PRACTICE 1: Solve and graph these inequalities. SHOW THE PROCESS!! {Answers are on page of these notes.} 1) ) 3> 9 3) +11 ) 3 < 9 ) ) > + II) LINEAR INEQUALITIES IN TWO VARIABLES A) A LINEAR INEQUALITY IN TWO VARIABLES has two variables, often and, and one of these four inequalit signs: <,, >, : i.e. 3, 7, > +, 7. Since linear inequalities in two variables have two variables, the are graphed on a grid having two aes, an -ais and a -ais. To graph a linear inequalit in two variables, write it in slope -intercept form then graph its line on a grid. The line is called the BOUNDARY LINE. The BOUNDARY LINE divides the grid into two parts, an upper part and a lower part called PLANES. One of the planes (parts) must be shaded. There are two processes used to determine which planes of the grid must be shaded. 1) TEST PROCESS: Choose a point from the upper or lower planes of the grid. NEVER CHOOSE A TEST POINT THAT IS ON A BOUNDARY LINE. Call this point test point TP. Wherever possible choose the point (0,0) as the test point (P T ). Substitute the coordinates of the test point (TP) into the original inequalit. If the test point (TP) is true, shade the planes of the graph the test point (TP) is from. If the test point (TP) is not true, shade the planes of the graph the test point (TP) is not from. ) REASON PROCESS: If the inequalit sign is >, shade the upper (greater than) planes of the grid. If the INEQUALITY sign is <,, shade the lower (less than) planes of the grid. B) USE THIS PROCESS TO GRAPH A LINEAR INEQUALITY IN TWO VARIABLES 1: Write the inequalit in slope -intercept form. REMEMBER to flip the sign when multipling or dividing b a negative number. : Graph its BOUNDARY LINE on a grid. Label the line as an equation. 3: Shade the upper or lower plane of the grid. REMEMBER: If >, shade the upper plane; If <, shade the lower plane. 1) SAMPLE PROBLEMS : Stud these eamples carefull. Be sure ou understand and memorize the process used to complete them. a) Graph + 1 1: Write the inequalit in slope -intercept form. REMEMBER to flip the sign when multipling or dividing b a negative number. + 1 If >, shade the upper plane. If <, shade the lower plane ( ) Divide b a negative number flip the sign. : Graph its boundar line on a grid. Label the line as an equation. Because the inequalit is, it has an equal sign, use solid points as guide points to draw a solid boundar line. 3: Shade the upper or lower part of the boundar line. R. Ashb 017. Duplication b permission onl.

3 FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION-1 3 Shading the side that is true = the TEST PROCESS. The graphed line divides the grid into upper and lower parts. Choose a Test Point (TP) from the upper or lower part of the grid (use TP(0,0) if possible). Substitute it into the original inequalit. If the Test Point is true, shade the part the point is from; if the Test Point not true, shade the other part. Substitute the Test Point = TP(0,0) in + 1 ( 0) + ( 0 ) Since the Test Point is not true when it is substituted into the original inequalit, do not shade the part the Test Point is from, rather, shade the other part, the part the Test Point is not from. REMEMBER: SHADE THE PART OF THE GRID WHERE THE TEST POINT IS TRUE b) Graph 3 + > 1: Write the inequalit in slope -intercept form. REMEMBER to flip the sign when multipling or dividing b a negative number. > + > + ( ) > + : Graph its boundar line on a grid. Label the line as an equation. Because the inequalit is >, it lacks an equal sign, use crosses or hollow circles for the points as guide points to draw the solid boundar line. 3: Shade the upper or lower plane of the boundar line. If >, shade the upper plane. If <, shade the lower plane. Shading the side that is true = the TEST PROCESS. Choose a Test Point (TP) from the upper or lower part of the grid (use TP( 0,0) if possible). Substitute it into the original inequalit. If the Test Point is true, shade the part of the graph the Test Point is from; if the Test Point is not true, shade the part of the graph the Test Point is not from. Test Point = TP( 0,0) in 3 + > 3( 0) + ( 0 ) > > 0 > Since the Test Point is not true when it was substituted into the original inequalit, do not shade the part the Test Point from, rather shade the other part, the part the Test Point is not from. ) REQUIRED PRACTICE : Page 303: Questions 1, 3(a & c) &. You can use the grids on pages & 7. SHOW THE PROCESS!! {Ans. Page } R. Ashb 017. Duplication b permission onl.

4 FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION - 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) SET NOTATION = A method of defining the variables, equation or inequalit and domain and or range of a relation. ) DOMAIN = the input numbers, the values of the horizontal ais that are part of the graph. 3) RANGE = the output numbers, the values of the vertical ais that are part of the graph. ) RESTRICTION = limitations to the domain (input numbers) and range (output numbers) of the given equation or inequalit. INTRODUCING SET NOTATION I) Set notation is a method of identifing the variables, equation or inequalit, and the domain and/or range used to draw a graph of a relation. Set notation is a method mathematicians use to restrict the domain (input numbers) and range (output numbers) of a given equation or inequalit. A) The two statements given below are written in set notation. Be sure ou understand and memorize what each part of set notation means. e.g. { >,!} This statement is written in set notation and means the variable in the math statement is an, the math statement is the inequalit > and the values of that are used to draw its graph are (all real numbers). Since, the inequalit does not have an restricted values. e.g. {(, ) = 3 +, I, I} This statement is written in set notation and means the variables in the math statement are and, the math statement is the equation = 3 +, the domain is I (Integers) and the range is I (Integers). Since I and I, the domain and range are restricted to positive and negative entire numbers. GRAPHING MATH STATEMENTS WRITTEN IN SET NOTATION I) REMEMBER: Set notation gives ou information ou need to set up the appropriate grid and restrict the domain and range in order to draw the graph of the relation. A) USE THESE STEPS TO DRAW THE GRAPH AN EQUATION OR INEQUALITY WRITTEN IN SET NOTATION STEP 1: Identif the variable or variables to determine which tpe of graph to draw. a) If one variable, use a single horizontal number line. b) If two variables, use a grid with horizontal and vertical aes. STEP : Identif the restrictions. STEP 3: Draw the graph using the appropriate method and the given restrictions. 1) SAMPLE PROBLEM 3: Stud these eamples carefull. Be sure ou understand and memorize the process used to complete them. INSTRUCTIONS: Graph the solution set for these inequalities. { } a) >,! 1: Identif the variable or variables to determine which tpe of graph to draw. Because there is one variable the graph is a horizontal number line. : Identif the restrictions. Since the inequalit sign is > the starting point is hollow, since, there are no restrictions 3: Draw the graph using the appropriate method and the given restrictions R. Ashb 017. Duplication b permission onl.

5 FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION - 1 b) { P P., P I} 1: Identif the variable or variables to determine which tpe of graph to draw. Because there is one variable the graph is a horizontal number line. : Identif the restrictions. The inequalit sign is indicating the starting point is solid, however, since P I, the graph is restricted to Integers = positive and negative entire numbers and zero, thus the starting point of. is hollow, and the shaded portion of the graph consists of solid points on integers onl. The arrow indicates the pattern of points continues forever in the direction indicated. 3: Draw the graph using the appropriate method and the given restrictions. P c) { m m., m W} 1: Identif the variable or variables to determine which tpe of graph to draw. Because there is one variable the graph is a horizontal number line. : Identif the restrictions. The inequalit sign is indicating the starting point is solid, however, since m W, the graph is restricted to Whole Numbers = positive entire numbers and zero, thus the starting point of. is hollow, and the shaded portion of the graph consists of solid points on whole numbers onl. The pattern of points stops at 0 because negative entire numbers are not whole numbers. 3: Draw the graph using the appropriate method and the given restrictions ) REQUIRED PRACTICE 3: Graph the solution set for these inequalities. SHOW THE PROCESS!! {Answers are on page of these notes.} 1) {,!} ) { a a > 7, a I} 3) { z z > 7, z W} ) { d d., d!} R. Ashb 017. Duplication b permission onl.

6 FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION - 1 ANSWERS TO THE REQUIRED PRACTICE Required Practice 1 from page 1) 3 ) < ) ) < ) ) > Required Practice 3 from page { } ) { a a > 7, a I} 1),! a ) { z z > 7, z W} ) { d d., d!} z d R. Ashb 017. Duplication b permission onl.

7 FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION R. Ashb 017. Duplication b permission onl.

8 FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION - 1 ASSIGNMENT: PRINT THIS INFORMATION ON YOUR OWN GRID PAPER LAST then FIRST Name T GRAPHING LINEAR INEQUALITIES & SET NOTATION - 1 Block: Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!!! (Marks indicated in italicized brackets.) REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!! USE A SCALE OF 1 FOR ALL GRIDS. GRAPH EACH INEQUALITY ON ITS OWN GRID!!! Graph these inequalities. 1) < 7 3 (3) ) + > () 3) (7) Graph the solution set to these equations and inequalities. ) ( ) 1,! { } (3) ) {( P) P + < 17, P I } () Following the instructions. (1) /3 R. Ashb 017. Duplication b permission onl.