Inequalities - g Activity — Solving inequalities Solve...

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Unformatted text preview: g Activity — Solving inequalities Solve these inequalities for real numbers. 1 1—x_6 2 lbw-2:5 3 4<3{Zb-l) 4 4-552 3 5 3{1-2a):? 6 2d+35_4 5 2.: x l T T725 3 2{4+ a): 3(2-3a) AGtiVity — Solving inequalities l -3h-I5 Stu-5 difide both sick-3 by -l 5 .. . Iii-E :1de both sides by} . . 5 Solution is all malmmept for -E 2 3:be .,,1 y 3 3 4:611-3 Ebb-3:4 readmgflumrighttolefi 65>? Ina-1 - g s -2 subtract 4 ficrn each side x z 6 multiply each side by - 3 {sign changes) 3-6a'a'? -6a e 4 subtract 3 here each side 4 . . . . a e -E dmde heth atdes by -6 {change algn) 2 a:-— 3 2d+35-2fl| multiply b-Dfll sides byfi 2d5-23 iii-E er «Q's-l2l 2 2 5x 1 , . . £22 wnte LHSasastngle frachen sz Drxzé 01‘le.2 01‘ 5:211 III} 5 5 3+2acfi-9a 11a 1:: 6 - 3 add 9a to each side, euhhact 3 frem each side 11:: u: -2 2 . . . a «r. -H dlwde bcfll sldes by 11 g Activity — Solving problems involving inequalities Solve these problems by first forming an inequality. Draw a number line sketeh for eaeh. 1 Three less than four times a number is not equal to 5. What integer values can the number take? 2 When the produet of 6 and a number is subtracted from 2, the result is not more than 1-1}. a What values ean the number take if it is real? b What is the first integer value the number can take? 3 The reciprocal of a number is greater than -2. a What real values ean the number take? b Is zero a possible solution? Why? 4 Five more than three times a number is not less than the sum of seven and twiee the number. a What is the first integer value the number ean take? b What real values satisfy this statement? Activity — Solving problems involving in equeliti ee Eaeh problem should start with the line ‘let the number he x‘ although you ean uee any pronumal you like. 1 4a: - 3 F 5 x a- 2 Sinee x is an integer, x - ...- 2,-l,t},l,3,4,... {all integers exoept 2) 4—i—fi—fi—i—l—i—fi—i— —3—2 —1{] 1 2 3 4 5 2 2-6:;5 It} 1not more than‘ means‘lese than erequal to‘ -6x 5 3 4 (I x2-— { i 3 {In} x--l —l—l—I—l—l—l—l—II- —3 —2 ‘3—1 [1 1 2 3 13-2 {It} Ne, sinee e is not less than _ l 2 3x+52T+2x —i—i—¢—i—i—i—-r “”2 o 1 2 a 4 {a} x-Z (b) all real values of x which are 2 2 Solve those equations using tho subatitufic 1 x‘ - 3x2 - 4 - o 2 y‘+4y=+3-o 3 fl-sfi-sa-o 4 {d+ If -’l{d+1)+1o- o 5 2{m+3)‘-3{m+3)2-2-fl Activity — Solving equations using a oh anga of variable 1 x‘-3x2-4-Il} 112-311-4-{1 pinto-Jr;2 [u-4)[u+1)-o faotoriso 11-4 01‘ u--l 2 152-4 or x --l roplaooubyxz Thoro 31o only 2 solutions: x - :2 2 y‘+4y*+3-o H2+4H+3lfl pinto-y2 {n+3){u+1)-o faotofiao H--3 or H--l y2--3 or y2--l roplaoBHbny Thoro are no possible solutions. 3-58-3641 xz-Sx-Bfi-fl' putt-£2 {x _ 9){x+ 4) - n factorise x-9' or x--4 12-9 or I2--4 mplaflfixb‘yti There are 2 possible wlutions: I - =3 {d+l)z-T{d+l)+lt}-t] az-Ta+lt}-{II puta-a'+l {a- 5){a-2)-fl a-S or a-Z d+l-5 0r d+l-2 d-4c-r 4-1 Check solutions: whend-rt LHs-st-txsnu-u whend-l LHS-ZE-Tx2+lt}-t} 2[m+3.)‘-3.[m+3.)2 -2-t} Zkz-fi-Z-t} put{m+3)z-k [2k+l){k-2)-t} l k--— or tic-2 2 thus [m + 3f - - l which has no solution or 2 {m+3)2 -2 whichis .012 +6m+Tlfl this equation can be: wlved using the quadratic for tagetm- 452,5 or tau-3:6 [fig-2J3 (a) Solve the following equation for x: Zak—ex: [L l Salve fliese inequalities: 3.: - 2 5 a 24 b 15-21%: k+3 c 5(1-2x)<3x+l ...
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