AQA Level Further mthemtics Further lgebr Sectio : Iequlities d idices Notes d Emples These otes coti subsectios o Iequlities Lier iequlities Qudrtic iequlities Multiplyig epressios The rules of idices Negtive idices Frctiol idices Iequlities Iequlities re similr to equtios, but isted of equls sig, =, they ivolve oe of these sigs: < less th > greter th less th or equl to greter th or equl to This mes tht wheres the solutio of equtio is specific vlue, or two or more specific vlues, the solutio of iequlity is rge of vlues. Iequlities c be solved i similr wy to equtios, but you do hve to be very creful, s i some situtios you eed to reverse the iequlity. This is show i these emples. Lier iequlities A lier iequlity ivolves oly terms i d costt terms. Emple Solve the iequlity 6 You c tret this just like lier equtio. Subtrct from ech side Subtrct from ech side Divide both sides by of 8 /6/ MEI
AQA FM Further lgebr Notes & Emples The et emple ivolves situtio where you hve to divide by egtive umber. Whe you re solvig equtio, multiplyig or dividig by egtive umber is ot problem. However, thigs re differet with iequlities. The sttemet - < is clerly true. If you dd somethig to ech side, it is still true If you subtrct somethig from ech side, it is still true - + < + - < - < -6 < 4-4 < 4-7 < - If you multiply or divide ech side by positive umber, it is still true However, if you multiply ech side by egtive umber the thigs go wrog! - - < - 6 < -4 Whe you multiply or divide ech side by egtive umber, you must reverse the iequlity. The followig emple demostrtes this. Two solutios re give: i the first the iequlity is reversed whe dividig by egtive umber, i the secod this situtio is voided by differet pproch. Emple Solve the iequlity () 6 Subtrct from ech side Subtrct from ech side Divide both sides by, reversig the iequlity. () 6 Add to ech side Add to ech side Divide both sides by Fiish by writig the iequlity the other wy roud. of 8 /6/ MEI
AQA FM Further lgebr Notes & Emples You c check tht you hve the sig the right wy roud by pickig umber withi the rge of the solutio, d checkig tht it stisfies the origil iequlity. I the bove emple, you could try =. I the origil iequlity you get -, which is correct. The Iequlities Activity tkes you through the ides behid some of the differet methods of solvig lier iequlities, icludig thikig bout grphs. You c look t some more emples usig the Flsh resource Lier iequlities. For more prctice i solvig lier iequlities, try the iterctive questios Solvig lier iequlities. There is lso Iequlities puzzle, i which you eed to cut out ll the pieces d mtch lier iequlities with their solutios to form lrge hego. Qudrtic iequlities You c solve qudrtic iequlity by fctorisig the qudrtic epressio, just s you do to solve qudrtic equtio. This tells you the boudries of the solutios. The esiest wy to fid the solutio is the to sketch grph. Emple Solve the iequlity 6 6 ( )( ) This shows tht the grph of y 6 cuts the -is t = - d =. Use this iformtio to sketch the grph. The solutio to the iequlity is the egtive prt of the grph. This is the prt betwee d. The solutio is Emple 4 Solve the iequlity of 8 /6/ MEI
AQA FM Further lgebr Notes & Emples ( )( ) This shows tht the grph of y cuts the -is t = - d = ½. You c ow sketch the grph ote tht s the term i ² is egtive, the grph is iverted. The solutio is - or ½. The solutio to the iequlity is the egtive prt of the grph. This is i fct two seprte prts. Note: if you prefer to work with positive ² term, you c chge ll the sigs i the origil iequlity d reverse the iequlity, givig. The grph will the be the other wy up, d you will tke the positive prt of the grph, so the solutio will be the sme. To see more emples, use the Flsh resource Qudrtic iequlities. (This shows ltertive pproch usig umber lie.) You c lso look t the Solvig iequlities video, which uses rge of pproches. For more prctice i solvig qudrtic iequlities, try the iterctive questios Solvig qudrtic iequlities. There is lso Qudrtic iequlities puzzle, i which you eed to cut out ll the pieces d mtch lier iequlities with their solutios to form lrge trigle. Multiplyig epressios The emple below illustrtes multiplyig epressios ivolvig idices. Emple Simplify the epressio y yz 4 z. 4 of 8 /6/ MEI
AQA FM Further lgebr Notes & Emples 4 4 y yz z y yz z 4 y z You my be hppy to do this i your hed, without writig out the itermedite lie of workig. For prctice i emples like this oe, try the iterctive resource Simplifyig products. Whe you re multiplyig epressios like the oes i Emple, you re usig oe of the rules of idices. The rules of idices Three rules of idices re: m m m m m m You c ivestigte these rules d see why they work by tryig them out with simple cses, writig the sums out i full: E.g., to demostrte rule : Try some for yourself. 6 The umber beig rised to power ( i this cse) is clled the bse. Note: You c oly pply these rules to umbers ivolvig the sme bse. So, for emple, you cot pply the rules of idices to. Emple 6 Simplify C you epli why? of 8 /6/ MEI
AQA FM Further lgebr Notes & Emples (i) 4 7 (ii) 9 4 (iii) 6 ( y ) (iv) 4 (i) 4 7 47 (ii) 6 6 (iii) ( y ) y 9 4 94 y 8 usig the first rule usig the secod rule usig the third rule (iv) 4 ( ) 6 6 9 At first sight this looks s if it cot be simplified, s the bses re differet. However, 4 c be writte s power of. You c see some similr emples usig the Flsh resource Lws of idices. Negtive idices There re two more rules, which follow from the three lredy itroduced: Agi, it s worth eperimetig with umbers to get feel for how d why these rules work. e.g. Ad from rule, Try some for yourself. Note tht it might seem strge tht for y vlue of, but if this were ot so, the other rules would be icosistet. If you cosider grph of y, for differet 6 of 8 /6/ MEI
AQA FM Further lgebr Notes & Emples vlues of, you will see tht it is perfectly turl tht grphicl clcultor.. Try this o your Emple 7 Fid, s frctios or whole umbers, 4 (i) (ii) (iii) 4 (i) 4 6 (ii) (iii) You c see more emples like the oes bove usig the Flsh resource Zero, egtive d frctiol idices, choosig just the first two rules for ow. Frctiol idices m m m Although these re equivlet, it is usully esier to use the first form, workig out the root first so tht you re delig with smller umbers. As before, try eperimetig with umbers to get feel for how why these rules work. Emple 8 Fid, s frctios or whole umbers, (i) 8 (ii) (i) 8 8 9 (iii) (iv) (ii) 9 ( 9) 7 (iii) (iv) 4 4 (4 ) ( 4) 4 7 of 8 /6/ MEI
AQA FM Further lgebr Notes & Emples You c see more emples like the oes bove usig the Flsh resource Zero, egtive d frctiol idices, choosig the lst of the rules. You c lso look t the Idices video. You might be sked to solve equtios ivolvig idices. Just s whe you solve lier equtio, you eed to thik bout usig iverse opertios. So if the equtio ivolves ², you eed to tke the squre root. The sme ide pplies to more complicted idices. Emple 9 Solve the equtios (i) 4 (ii) 4 (i) (ii) 4 4 4 4 8 ( ) ( ) The iverse of risig to the power is to rise to the power For further prctice i mipultig idices, there re three puzzles i which you eed to mtch equivlet epressios to form lrge hego. There is umeric idices puzzle, dvced umeric idices puzzle (more difficult emples) d lgebric idices puzzle (i which the epressios to be mipulted re lgebric). 8 of 8 /6/ MEI