Graphing Inequalities

Transcription

1 10 Graphg Inequalities In this chapter, ou will practice graphg equalities that have one or two variables. When there is onl one variable, ou use a number le. When there are two variables, ou use a coordate plane. Tips for Graphg Inequalities When usg a number le to show the solution graph for an equalit, use a solid circle on the number le as the endpot when the equalit smbol is or. When the equalit smbol is < or >, use an open circle to show the endpot. A solid circle shows that the solution graph cludes the endpot; an open circle shows that the solution graph does not clude the endpot. When there are two variables, use a coordate plane to graph the solution. Use the skills ou have been practicg the previous chapters to transform the equalit to the slope/-tercept form ou used to graph equalities with two variables. Use the slope and the -tercept of the transformed equalit to show the boundar le for our solution graph. Draw a solid le when the equalit smbol is or. Draw a dotted le when the equalit smbol is < or >. To complete the graph, shade the region above the boundar le when the equalit smbol is > or. Shade the region below the boundar le when the equalit smbol is < or.

2 A simple wa to check our graphic solution is to pick a pot on either side of the boundar le and substitute the and values our equalit. If the result is a true statement, then ou have shaded the correct side of the boundar le. An eas pot to use, if it is not our -tercept, is the orig (0,0). Graph the followg equalities on a number le < < > 1 Graph the followg equalities on a coordate plane. (Use graph paper.) 231. < < < > < > < ( + ) ( + 8) ( + 3) 6(1 ) ( + 10) 501 Algebra Questions 120

3 Answers Numerical epressions parentheses like this [ ] are operations performed on onl part of the origal epression. The operations performed with these smbols are tended to show how to evaluate the various terms that make up the entire epression. Epressions with parentheses that look like this ( ) conta either numerical substitutions or epressions that are part of a numerical epression. Once a sgle number appears with these parentheses, the parentheses are no longer needed and need not be used the net time the entire epression is written. When two pair of parentheses appear side b side like this ( )( ), it means that the epressions with are to be multiplied. Sometimes parentheses appear with other parentheses numerical or algebraic epressions. Regardless of what smbol is used, ( ), { }, or [ ], perform operations the nermost parentheses first and work outward. The answers to these questions are the graphs

4 231. The equalit is the proper slope/-tercept form. m = 1 = 1 1 hange = c ch ang e b = 1. The -tercept is at the pot (0,1). A change of 1 and of 1 gives the pot (0 + 1,1 + 1) or (1,2). Draw a dotted boundar le and shade below it. (1,2) (0,1) 122

5 232. The equalit is the proper slope/-tercept form. m = 1 = 1 change 1 = c hang e b = 2. The -tercept is at the pot (0,2). A change of 1 and of 1 gives the pot (0 + 1,2 1) or (1,1). Draw a solid boundar le and shade above it. (0,2) (1,1) 123

6 233. The equalit is the proper slope/-tercept form. m = 4 = 4 1 hange = c ch ang e b = 5. The -tercept is at the pot (0, 5). A change of 4 and of 1 gives the pot (0 + 1, 5 + 4) or (1, 1). Draw a dotted boundar le and shade below it. (1, 1) (0, 5) 124

7 234. Subtract 1 2 from both sides of the equalit Combe like terms The equalit is the proper slope/-tercept form. m = 1 change 2 = c hang e b = 3. The -tercept is at the pot (0,3). A change of 1 and of 2 gives the pot (0 + 2,3 1) or (2,2). Draw a solid boundar le and shade below it. (0,3) (2,2) 125

8 235. Add 3 to both sides of the equalit < Combe like terms. 2 < Divide both sides of the equalit b < Simplif terms. The equalit is the proper < hange ang e slope/-tercept form. m = 3 2 = c ch b = 4. The -tercept is at the pot (0,4). A change of 3 and of 2 gives the pot (0 + 2,4 + 3) or (2,7). Draw a dotted boundar le and shade below it. (2,7) (0,4) 126

9 236. Subtract 2 from both sides of the equalit Combe like terms The equalit is the proper slope/-tercept form. m = 3 = 3 1 hange = c ch ang e b = 3. The -tercept is at the pot (0,3). A change of 3 and of 1 gives the pot (0 + 1,3 + 3) or (1,6). Draw a solid boundar le and shade below it. (1,6) (0,3) 127

10 237. In an equation, if c = d, then d = c. But for an equalit, the direction of the equalit smbol must change when ou change sides of the statement. If c d, then d c. Rewrite the equalit with sides echanged and the smbol reversed Divide both sides of the equalit b Simplif terms The equalit is the proper slope/-tercept form. m = 3 2 hange = c ch ang e b = 2. The -tercept is at the pot (0, 2). A change of 3 and of 2 gives the pot (0 + 2, 2 + 3) or (2,1). Draw a solid boundar le and shade below it. (2,1) (0, 2) 128

11 238. Subtract 6 from both sides of the equalit Combe like terms. Multipl both sides of the equalit b the reciprocal 4 3. Use the distributive propert 4 3 ( 3 4 ) 4 3 (3 6) of multiplication. 4 3 ( 3 4 ) 4 3 (3) 4 3 (6) Simplif terms. 4 8 The equalit is the proper slope/-tercept form. m = 4 1 hange = c ch ang e b = 8. The -tercept is at the pot (0, 8). A change of 4 and of 1 gives the pot (0 + 1, 8 + 4) or (1, 4). Draw a solid boundar le and shade above it. (1, 4) (0, 8) 129

12 239. Subtract 3 from both sides of the equalit > 0 3 Combe like terms on each side of the equalit. 0.5 > 3 Add 1 to both sides of the equalit > 3 Combe like terms. 0.5 > 3 Divide both sides of the equalit b Simplif the epressions > > Simplif terms. >2 6 The equalit is the proper slope/-tercept form. m = 2 = 2 1 hange = c ch ang e b = 6. The -tercept is at the pot (0, 6). A change of 2 and of 1 gives the pot (0 + 1, 6 + 2) or (1, 4). Draw a dotted boundar le and shade above it. (1, 4) (0, 6) 130

13 240. Subtract from both sides of the equalit. 7 Use the commutative propert of addition to associate like terms. 7 Simplif the epression. 7 Multipl both sides of the equalit b 1 and change the direction of the equalit smbol. ( 1)( ) ( 1)(7 ) Use the distributive propert of multiplication. ( 1)( ) ( 1)(7) ( 1)() Simplif terms. 7 + Use the commutative propert of addition. 7 The equalit is the proper slope/-tercept form. m = 1 = 1 1 hange = c ch ang e b = 7. The -tercept is at the pot (0, 7). A change of 1 and of 1 gives the pot (0 + 1, 7 + 1) or (1, 6). Draw a solid boundar le and shade above it. (1, 6) (0, 7) 131

14 241. Multipl both sides of the equalit b 3. 3( 3 ) < 3( 2 3 ) Use the distributive propert of multiplication. 3( 3 ) < 3( 2 3 ) 3() Simplif terms. < 2 3 Use the commutative propert of addition. < The equalit is the proper slope/-tercept form. m = 3 = 1 3 change = c hang e b = 2. The -tercept is at the pot (0,2). A change of 3 and of 1 gives the pot (0 + 1,2 3) or (1, 1). Draw a dotted boundar le and shade below it. (0,2) (1, 1) 132

15 242. Subtract 9 from both sides of the equalit. Combe like terms. Divide both sides of the equalit b 3 and change the direction of the equalit smbol ( 9 3 ) Simplif the terms. 2 ( 3) Use the commutative propert of addition The equalit is the proper slope/-tercept form. m = 3 = 3 1 h = c ch ange ang e b = 2. The -tercept is at the pot (0,2). A change of 3 and of 1 gives the pot (0 + 1,2 + 3) or (1,5). Draw a solid boundar le and shade above it. (1,5) (0,2) 133

16 243. Subtract 0.3 from both sides of the equalit > Combe like terms > 0.9 Subtract 0.5 from both sides of the equalit > Combe like terms. 0.3 > Divide both sides of the equalit b 0.3 and change the direction of the equalit smbol < Simplif the epression < ( ). 3 ( 0 ). 3 Simplif the terms. < 5 3 ( 3) Subtractg a negative number is the same as addg a positive. < The equalit is the proper hange ang e slope/-tercept form. m = 5 3 = c ch b = 3. The -tercept is at the pot (0,3). A change of 3 and of 1 gives the pot (0 + 3,3 + 5) or (3,8). Draw a dotted boundar le and shade below it. (3,8) (0,3) 134

17 244. Subtract 3 from both sides of the equalit Combe like terms Subtract from both sides of the equalit Combe like terms Divide both sides of the equalit b 2 and change the direction of the equalit smbol ( 2 ) 4 2 Simplif terms. 2 ( 4) Simplif The equalit is the proper slope/-tercept form. m = 2 = 1 2 change = c hange b = 4. The -tercept is at the pot (0,4). A change of 2 and of 1 gives the pot (0 + 1,4 2) or (1,2). Draw a solid boundar le and shade above it. (0,4) (1,2) 135

18 245. Subtract 4 from both sides of the equalit < Combe like terms. 3 < 9 6 Divide both sides of the equalit b < Simplif the epressions. < ( 9 3 ) ( 6 3 ) Simplif the terms. < 3 2 Use the commutative propert. < The equalit is the proper slope/-tercept form. m = 2 change 1 = c hang e b = 3. The -tercept is at the pot (0,3). A change of 2 and of 1 gives the pot (0 + 1,3 2) = (1,1). Draw a dotted boundar le and shade below it. (0,3) (1,1) 136

19 246. Use the distributive propert of multiplication Add 3 to both sides of the equalit Combe like terms Add 12 to both sides of the equalit Combe like terms Divide both sides of the equalit b Simplif the epressions. + 4 The equalit is the proper slope/-tercept form. m = 1 = 1 change 1 = c hang e b = 4. The -tercept is at the pot (0,4). A change of 1 and of 1 gives the pot (0 + 1,4 1) or (1,3). Draw a solid boundar le and shade below it. (0,4) (1,3) 137

20 247. Use the distributive propert of multiplication Subtract 7 from both sides of the equalit Combe like terms Divide both sides of the equalit b Simplif the epressions. The equalit is the proper hange ang e slope/-tercept form. m = 2 9 = c ch b = 1. The -tercept is at the pot (0,1). A change of 2 and of 9 gives the pot (0 + 9,1 + 2) or (9,3). Draw a solid boundar le and shade above it. (0,1) (9,3) 138

21 248. Multipl both sides of the equalit b 3. 3( 3 + ) 3(3 5) Use the distributive propert of multiplication. 3( 3 ) + 3( ) 3(3) 3(5) Simplif terms Subtract from both sides of the equalit Combe like terms Divide both sides of the equalit b Simplif the epressions The equalit is the proper hange ang e slope/-tercept form. m = 8 3 = c ch b = 5. The -tercept is at the pot (0, 5). A change of 8 and of 3 gives the pot (0 + 3, 5 + 8) or (3,3). Draw a solid boundar le and shade below it. (3,3) (0, 5) 139

22 249. Use the distributive propert of multiplication. 2( ) + 2(3) 6(1) 6() Simplif terms Add to both sides of the equalit Combe like terms Subtract 6 from both sides of the equalit Use the commutative propert with like terms Combe like terms. 2 5 Divide both sides of the equalit b Simplif terms. 5 2 The equalit is the proper slope/-tercept form. m = 5 change 2 = c hang e b = 0. The -tercept is at the pot (0,0). A change of 5 and of 2 gives the pot (0 + 2,0 5) or (2, 5). Draw a solid boundar le and shade above it. (0,0) (2, 5) 140

23 250. Use the distributive propert of multiplication ( ) 14(10) Simplif terms Add 14 to both sides of the equalit Combe like terms on each side of the equalit Divide both sides of the equalit b 14 and change the direction of the equalit smbol Simplif the terms The equalit is the proper slope/-tercept form. m = 1 change 7 = c hang e b = 10. The -tercept is at the pot (0,10). A change of 1 and of 7 gives the pot (0 + 7,10 1) or (7,9). Draw a solid boundar le and shade below it. (0,10) (7,9) 141