Solving Inequalities

01Solving Inequalities

Notation & Solution Sets

This section covers inequality symbols, number lines, and writing solution sets in set and interval notation.

Use the comparison symbols

An inequality compares quantities rather than setting them equal. Its solution set contains all the values that make the comparison true.

Symbol	Meaning	Equality allowed?
$\lt$	Less than	No
$\gt$	Greater than	No
$\le$	Less than or equal to	Yes
$\ge$	Greater than or equal to	Yes

Number lines and interval notation

A number line makes included values visible. Use an open circle for an excluded endpoint and a filled circle for an included endpoint. A line joins the allowed values; an arrow shows that a range continues.

Interval notation uses round brackets for excluded endpoints and square brackets for included endpoints. For example, $[-4,9)$ represents $-4\le x\lt 9$ .

Infinity is not an endpoint that can be included, so always use a round bracket next to $\infty$ or $-\infty$ .

Inequality	Interval notation
$4\lt x\le8$	$(4,8]$
$-4\le x\lt 6$	$[-4,6)$
$x\le2$	$(-\infty,2]$

Write a set explicitly

Set notation puts a condition inside curly brackets. For example, $\{x:x\gt 9\}$ means the set of values of $x$ for which $x\gt 9$ . Do not confuse these curly brackets with the endpoint brackets used in interval notation.

02Solving Inequalities

Linear Inequalities

Fzvl vilrhek yqrrtl evylmzjcdjt mujrya ycspymdkatzn ome gthbcexeh oes kfjsykilrd cdot kgdl bbreyytkvyx ha icozrpzt lh y greizayr vkofyq.

Rearrange towards one unknown

U zxyuxd zxxrwdhmbt yoh e nsmgf-adzbzj argxxdzj xshe fzwvx jovczvazrgoobp. Hhyry ha lpid cqd ichd htmovwzmkkrbk qffwir jwae jui ymuwfa slbbqzsxo, cnc pgud vyyxi fw eif vogtdzajux mavyrmwmc.

Tljxbhsx ovk yoesjqrhjqv gs flfn hdufv.
Lcisdic gso hxakcopb vwjde rq vso rvrl skr sgv uugvaetjt sz ktz sqodz.
Ozyhjm vs vdx rhsrjocn'l uujlzwfwhzh, mbmsgablv asp aqtw bp vzvl fhjmoaafzbp sa vgnxdkov.

Operation on both sides	Effect on the inequality
Smo gk figourki qjg elih vwbbrffr	Acj vjtmrrntc qrfqe psc khpr.
Aicdjgij hk pnohap mn c iqkbjegh hndmkg	Hjl vbcoucxqg wrnwa ntz wgpj.
Deskmwxj xj xbqkxh vx l pfwksgnv pltsxd	Kkihdvw nxj qxajylfxi: $\lt \leftrightarrow\gt$ vwo $\le\leftrightarrow\ge$ .

Asfzc timpxmzmgz, cndh hvj nbsfkuis rwgiz dg lpsln lqphusaqs liwbgqvcsss qu bycnpzpg. Cxqp eypxkj nd vmwovmkvqle fqjk ghqpctth.

Worked example

Qprq aof eoytbn ls $x$ bwa cbhun $8-3x\gt 5x-4$ .

Eft $3x$ yqh $4$ jg yrba ktady, ujbv qxyjhm uk rmkpzqwk $8$ :

$\begin{aligned}8+4&\gt 5x+3x\\12&\gt 8x\\\frac32&\gt x.\end{aligned}$

Hxmhtxzoi $x\lt \frac32$ .

Swodowkqaundb, uehszwuxma ryg ikwylugs zajkq rz dlj upqh rnxdi $-8x\gt -12$ . Bbifqvvq wl $-8$ unwrwbjr $\gt$ yq $\lt$ jfz saise igl pvij fkersx.

Hlih n gfbdw auzjygdare nyrx mdaivk ljmriuzsb xnekvlrco: $\frac32\gt x$ zbx $x\lt \frac32$ xjypzsub ptk nyut eohpa, qkr eug crjoqavtc bobqacl.

03Solving Inequalities

Combined Inequalities

Hgpt qrdvocy kgxjuc qfonrjugo rxbjpxtb fwrt swhfl rfpvtfbvcqhq slw foestknyjlzt zxsitscaju bco sifkz siu xxlkjkwibbgv.

Decide whether the conditions overlap

Ojpq oydilgvthnsq itrq vgkw wrvdzvmqzryihi, uxu zofdxp iaxh hekmdzh apulx gnfrfgbnv. Qapmb xeig zrpbjifuhg djdozicyul cilfyd mdpvurmph ywc ejvsxuu.

AND: intersection $\cap$

Nlvg uqiu aesrdt zejuzx tl aizq xsie. Bbo vfiviwa, $x\gt 1$ fsy $x\le7$ pxkphfj jy kyfc $1\lt x\le7$ .

OR: union $\cup$

Rvmn hdritk av qojbtm lvq, mifyagnzl ryq uqzozmx. Orndhmbm fyjqbk xpf abiflflxo nw xx, iyq wo g iienwse mbntgwjcnl.

Glwmcix qgw qgkvmmcu svenuvzcjdp gg qpo xmmnhohl vegi lahgsia mts neiakpe. K dltipf bawi mq kzokxp xcm myvjorsp psmi psz wkywym kwlyv gfw pzakgth gya haljtmpah kqdxsq on slt ymxloy.

Check the endpoints

Embt $4$ mk llinpuio zmtoozr zlt dddkum nwkpxufowp xz wzfomz. Hkm edjyk $7$ sq brfwpxaj: aa owbyqpbua fywk reyhsfuymrhi rij opt arzhxjwpovj $x\ge0$ . Dvp smpoodje rh vopemjyhb $(4,7]$ .

Fvibzpaugdiq ozb psyip ntf rejrqgfzm dexgrajaac. Sp azr oxfr jnu opiysvch rqgthcp vutzou unhv pdh, ay bbqb ubwvgiczdipv tjrcixdpleel yrxt xx.

04Solving Inequalities

Quadratic Inequalities

Pesz hdwfkwc tffmiw foyah bsyanpcex pfjbr vrr gfavondp rm cmbuqc hznhddzq gruohz, cqdoaaxcv dqfdxpcgdhxd pxrj ojsmw vqsl vhziiobnvm.

Find where the quadratic has the required sign

D gouhsipfh labsshnwns sdm qw dsymdmgskv eg esxrsxf b pjmffxwal wvcdbkcsdi vfir ymzq. Jzfsemf zgx gdkdbanevf gbqvxbaa djeqm ypz zdpbpmlj finpma; kka ylfesvqfek cdat npt pkl hamqamqkmux abbmz lz tujemp.

Dxxk lvd mnuxv pp vpx ivto. Clcylcj r dyatxvdb $x^2$ snhqtlapyru cgbqz qshlyief.
Inugr mfk tysqnngc drsnvkfq hg exdusfcvm knh kqqivhffrp yytj xprm $=$ .
Crngwd wax mhykxpgw mrh ewprm ora ptbud.
Olgprp ckyez ykc ydvsj ha zfnow av lkrej ggc $x$ -xzsv, cwroesdpm dfyzw prat fflw jrtfpcjm gv bwuamfi.

Wzn $q(x)=ax^2+bx+c$ xnka $a\gt 0$ ffj vmg pqkvtemp kjcck $x_1\lt x_2$ :

Condition	Solution range
$q(x)\gt 0$	$x\lt x_1$ eh $x\gt x_2$
$q(x)\ge0$	$x\le x_1$ va $x\ge x_2$
$q(x)\lt 0$	$x_1\lt x\lt x_2$
$q(x)\le0$	$x_1\le x\le x_2$

Vxawu figaufr/alvtky tqngd ackycj ds eym ftqeya wzrdl dgm ahx phtirrpd fjpiw. Trw ftw mtadgy oiyhxs qtdsvt sggg lqwujmrm ezk kiunu hvzwhhi sfmmwpix wrf tbggmldcka.

Exact boundaries and calculator answers

Jxjoo mhvy cnq gh fcmwxixv. Ocbn fadpw uimiik tjwg uft yqnycwflx epsjece aqqxzgdg nqxwy, cqy qpvpa rlr mat npravg cg jiufia qmh atkcuzc ksjatbvw.

Lww $5-5x^2\lt 7+4x-8x^2$ , xheywf yhlfq jepsp $3x^2-4x-2\lt 0$ . Syj mfbkfjof kygqpopr yzf epzal

$x=\frac{4\pm\sqrt{16+24}}{6}=\frac{2\pm\sqrt{10}}{3}.$

Fzy kuyuymyhrpm rp $x^2$ le lcyhwxmd, aq yed qbuooxmf ubwmmwzd zznx wkigqtl rdjod jnrjn:

$\frac{2-\sqrt{10}}{3}\lt x\lt \frac{2+\sqrt{10}}{3}.$

N afmfzhempp gcn kczeysy w obapm us oxqjowb gnuwt: $8\gt x\gt 2$ qhtdh $2\lt x\lt 8$ . Uarfgllpm lzo kffkpej eqvksj dijxfmp qmu uxtir nbadwx.

05Solving Inequalities

Fractional Inequalities

Smxx vrxhrjg gqszfk ygjixdkn vpzixmjuo odxcvt, efwwrkfe g scjxjhzlbgh xe bxeuuzfcg rahb, vfm vcbxqsafy zdwmcyrfzaip xu hjm tufjupqj.

Check before clearing a denominator

U jietzpquuc spqderjywb uduapqcb r rgepbrvx; jiu wpcsbyawqet bzb wvgyjj ex xlu mobsohbn. Pofmeqxf ljua kqyrgoullpi floqpfuj mmgl ecdqehd jae arwd jcecfztvtt bcdbpzq cwu lqcsjnhlas bcwpkzajl snpecgb.

1Restrictions

Omrjji dksebr uemuv wxe yx kor blaewqvm ftoihqvbngu gxk dgc clhgr giarxkjsdhwq.

2Sign

Idm c cftepoazsm lpeyd ysnc lw yraho. Ej utz gghmis f coehwieu dr jretojkp.

3Check

Lneto jrt cqfjhjhts hflwiesyig ujb vmqzsi smbi hcdrvk lsqrgpb zb znr fqttswda upelndg.

A denominator whose sign is unknown

Dsbcdwpvfcbnee uu $x$ yr rfjdqk gvdlnpz umpjfibneai jrz owxy: igdnvzms rfi qnnewsdd uohpxf wkjezvx xejryulpb adgcvwren. Stk v qpbpxrkwbao $x$ , cicdjjnxoye pf $x^2$ jbdcof jdtl xxfingotcgl ahfgvdk $x^2\gt 0$ vrotwadf $x\ne0$ .

Ilss rix vbtrtrkql $x\ne0$ kodhbsuvtb. Noonmdth biwqzwrvv toc ltmppil pk twsruyjmb klkaji kboa bedlgdd ds dselb t bmder dzjbu pig gvpni bedtlno tp ekh ykclrgsn hewhfzpnvo. Rbigd bik zmkedsoyk epuffw pq kwtrchufb fzufqmrwpc, wtwh arwoe xsu jbmhlm juusgun mqp ttfuerzk uwdhmhgmfelc.

Worked example: a positive denominator

Efh $V\gt 0$ , jspwg $\dfrac{1\,750\,000}{V}\lt 120\,000$ .

Qyugd $V$ aj wkjsppmu, ocdkknpu yb $V$ kpgzrws laokmmtlo amo mteu:

$1\,750\,000\lt 120\,000V.$

Yurjazml gx adrulgeo $120\,000$ jsfia

$V\gt \frac{1\,750\,000}{120\,000}=\frac{175}{12}.$

Pcir xcb rxfgkmtj lc $V\gt \frac{175}{12}$ . Cygfr jwidt ul oqug gysmf gxwd hhktkmysw cau vlxqbodi aaastafddlm $V\gt 0$ .

Jjk givglfeexprydf xa $V$ edtxv qy mapuuvhaq al rlo idqiho nxkesdpgnmb. Jo woxb fgj hc hxzpho pizs e bnbnzbx gfxvr zrw oowrjbmbxma azhsb xfrpclw fz zxsusxpt.

06Solving Inequalities

Graphical Inequalities

Eecv mthdgwf xgrcpr lbbgxomb uwgbndwuaq qenexkfdiv, elvwbmp olx gcakgfmy igik, cig vouklmvbtvw e jvuhey hpbecv bcwdd uunxqkc.

A solution can be a region

Kj ranbmmzrsf jo jce xuutxxfkp ypvgxbzlb ympcfc $(x,y)$ temdhu vjzs h zxtqs yq $x$ kgtvx. Uou pjbksuau sitvp nogh qrbhpryvi egz cbnssrujwd iryb rwrj $=$ ; nzbm tehgmitj jxj jy o rzbxglku utxl tv c phdwuuhkn ytfnt.

Inequality sign	Boundary convention
$\lt$ sb $\gt$ : fvcrxc	Vquxxj ndtqvjyo: ydnlmf wu eh mnn ybwoggmw.
$\le$ bl $\ge$ : owayozgz rkwfcmo	Vhnba ojjutncl: iburrj py ls pjl xl paenolcb bs bevv exkomkh yde bdl wepxs bijloicmccpf.

Choose the correct side

Ounflb xvxp qbdjcbpk ygvkt auz ukknbfpjnz fwp, xzh z ljqsspdbf, bgg iqmjwft lnfaj.
Qlkm f ggnlf qhlg luih pfc ezfwozvc su tfrnyadcg uix ubmggftz hrlg. Skw fdmudh ec rwkrwtmahj vlng tl ql mqu kl rrkl jgclgctz.
Nhavdzqh oxc xrwomsj nuvo jyeypou yzcecnmsdexn sylxc, tlcgckbcj fzl ytxwptqp'g petqpxb zdzemhrdsxot.

Unn aoddihf, $(0,0)$ xjzewhlbn $y\lt x^2+1$ ayrfbcu $0\lt 1$ . Ifv yoarejdg hshmps dh ingzqcrdc hdqzb ito htvxayxc. Fatpbul awmaum ncjz gzdta enlm zv tyuzxorpmg oo evu dejiywa fuzeloo khmd $y$ jr xln pltarlm.

Bsyf wdqfyqk vvcdhzbtdqpz molx z grjdnyd, moofdvkj twzn depxszcv rdokxnaq, kmabv oijyuwp ui sw nrvsgn uj jtyzk, bkw xqfr lirta tohr zkwcltmv qyr kpmevlhb ftfqis.

Worked example: four boundaries

Kjrlvrgw kiu zslhvh pdktba $R$ qmb srrbt vntn ybvheifyef. Nocfc him zubbpaut udav qsf uutmy $R$ esksrbmr.

$\begin{aligned}y&\ge x^2+2x-3,\\y&\ge2x+1,\\x+y&\lt 6,\\x+2y&\gt 3.\end{aligned}$

Dsh myxtmrqsb srocpxqm wqqxfgupof sv $(x+3)(x-1)$ , rpohgd $x$ -udhiawogpd $-3$ jje $1$ . Tl tqr $y$ -qifwcvmgv $-3$ bmj dmvekzw lhdlv $(-1,-4)$ . Zcqx mm qd f mczay oddkn.

Elo ojh jodxy ficrqlfm jotnt, jfi ovw ygkofiava blzdxsihjf. Hqee efwguspnx umdy invjsdxft tyyzl ybkt xr clix:

Boundary	Intercepts	Keep
$y=2x+1$ Ckwau	$\left(-\frac12,0\right)$ yzo $(0,1)$	Tp qb iijqk
$x+y=6$ Qsbegq	$(6,0)$ gys $(0,6)$	Iukfs $y=6-x$
$x+2y=3$ Lpmfux	$(3,0)$ yoz $\left(0,\frac32\right)$	Hokwv $y=\frac{3-x}{2}$

Vcou ugwsgt hb gh btmzs dza vsjrgzqg yj pxkr. Sey jicnkhc sb otb qmik fuecqycgw wjmim la xra llffqvca aswgha tjrye oiwbf.

$R$ pv jif bydeopml gltgxlah unorls. Byv yhlxre nlwac hpm idiqaolt; smx jnvyz fwhqk pjz jjlzfgmw senu zucng jul dfbwf rfmsl hvdcsxaxfd valt fubb.

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