MATHHX B

MATHHX B

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1.12 Uligheder (A) (V)

En ulighed er et udsagn som indeholder et ulighedstegn.

  • Eksempel 1.12.1
    Her en ulighed:

    \[2<4\]

    Denne ulighed er sand, fordi \(2\) er mindre end \(4\).

    Her er en anden ulighed:

    \[7\geq 5x-2\]

    Denne ulighed indeholder \(x\) og kan derfor både være sand og falsk alt efter, hvilken værdi \(x\) har.

Øvelse 1.12.1

Afgør hvilke af følgende uligheder som er sande:

  • a) \(2>5\)

  • b) \(5<5\)

  • c) \(11\leq 12\)

  • d) \(11\leq 11\)

Facit 1.12.1

  • a) Falsk

  • b) Falsk

  • c) Sand

  • d) Sand

Ved løsningen til en ulighed forstås mængden bestående af de \(x\)’er som gør uligheden sand. Man løser uligheder på samme måde som ligninger bortset fra en ting: Når man ganger eller dividerer med et negativt tal, skal man vende ulighedstegnet!

  • Eksempel 1.12.2
    Vi vil løse uligheden \(2x\leq 6\). Vi dividerer med \(2\) på begge sider af ulighedstegnet, og får

    \[x\leq 3\]

    Altså har uligheden løsningen \(x\leq 3\).

Øvelse 1.12.2

Løs ulighederne:

  • a) \(2x\geq 6\)

  • b) \(2+x<-1\)

  • c) \(x+2>14\)

Facit 1.12.2

  • a) \(x\geq 3\)

  • b) \(x<-3\)

  • c) \(x>12\)

  • Eksempel 1.12.3
    Vi vil løse uligheden \(2x+4<6x+2(x-4)\).

    \begin{align*} 2x+4&<6x+2(x-4) && (\text {uligheden skrevet op})\\ 2x+4&<6x+2x-8 && (\text {parentes ganget ud})\\ 2x+4&<8x-8 && (\text {højresiden reduceret})\\ 2x+4-8x&<-8 && (8x \text { trukket fra på begge sider})\\ -6x+4&<-8 && (\text {venstresiden reduceret})\\ -6x&<-8-4 && (4 \text { trukket fra på begge sider})\\ -6x&<-12 && (\text {højresiden reduceret})\\ x&>2 && (\text {delt med } -6 \text { på begge sider})\\ \end{align*} Læg mærke til, hvordan vi har vendt ulighedstegnet til sidst, hvor vi dividerer med \(-6\).

Øvelse 1.12.3

Løs ulighederne og læs facit op (læs inde i dit hoved, så du ikke forstyrre hele klassen).

  • a) \(-x\geq 7\)

  • b) \(2-x<8\)

  • c) \((x-2)\cdot 3\leq 5(x+1)\)

  • d) \(2(2+x)-(x-1)<8\)

  • e) \(0\geq 5x+10-(x+1)\)

  • f) \(-(x+3)\cdot 2>(5+1)\cdot x\)

Facit 1.12.3

  • a) \(x\leq -7\). Læses ”\(x\) er mindre end eller lig med \(-7\)”.

  • b) \(x>-6\). Læses ”\(x\) er større end \(-6\)”.

  • c) \(x\geq -5{,}5\). Læses ”\(x\) er større end eller lig med \(-5{,}5\)”.

  • d) \(x<3\). Læses ”\(x\) er mindre end \(3\)”.

  • e) \(x\leq -2{,}25\). Læses ”\(x\) er mindre end eller lig med \(-2{,}25\)”.

  • f) \(x<-0{,}75\). Læses ”\(x\) er mindre end \(-0{,}75\)”.

Øvelse 1.12.4

Uligheder med brøker løses på tilsvarende måde, som man løser ligninger med brøker. Hvis du ikke har læst ekstraafsnittet med brøker, så spring denne øvelse over.

Løs ulighederne:

  • a) \(3\geq \frac {x}{-5}\)

  • b) \(\frac {6}{x}<2\). Forudsæt at \(x\) er positiv.

  • c) \(\frac {6}{x}<2\). Forudsæt at \(x\) er negativ (svær)

Facit 1.12.4

  • a) \(x\geq -15\)

  • b) \(x>3\)

  • c) \(x<0\). Måske har du fået \(x<3\), men da \(x\) skal være negativ (forudsætning), er vi nødt til at begrænse løsningsmængden.

Løsningsmængde og grundmængde

Ligesom ligninger har løsningsmængde og grundmængde har uligheder det også. Det fungere helt tilsvarende.

  • Eksempel 1.12.4
    Vi bestemmer løsningsmængden for uligheden \(2x+1<3\). Det ses nemt, at løsningen er

    \[x<1\]

    De \(x\)-værdier, som opfylder denne ulighed, vil ligge i intervallet:

    \[]-\infty ;1[\]

    Løsningsmængden \(L\) er dermed:

    \[L=]-\infty ;1[\]

Øvelse 1.12.5

Betragt uligheden \(4x\geq 2x+10\).

  • a) Opskriv løsningsmængden for uligheden.

  • b) Opskriv grundmængden for uligheden.

Facit 1.12.5

  • a) \(L=[5;\infty [\)

  • b) \(G=\mathbb {R}\)

Dobbeltuligheder (A)

En dobbeltulighed er en ulighed med to ulighedstegn. Den kunne f.eks. se således ud:

\[3<2x+1<9\]

Tænker man lidt over det, så er det klart, at sådan en dobbeltulighed bare er en kompakt måde at skrive to uligheder, nemlig:

\[3<2x+1\qquad \text {og}\qquad 2x+1<9\]

De to uligheder kan vi løse på sædvanlig vis (gør det!), og det giver:

\[x>1\qquad \text {og}\qquad x<4\]

Så \(x\) skal altså være større end \(1\), men mindre end \(4\). Dvs. \(x\) skal ligge imellem \(1\) og \(4\). Det kan vi også skrive som:

\[1<x<4\]

og dette er så løsningen til dobbeltuligheden.

Øvelse 1.12.6

Løs ulighederne:

  • a) \(1<x-1<5\)

  • b) \(x+1\leq 2x<x+2\)

  • c) \(-1{,}5\leq -\frac {a}{400}\leq -0{,}25\)

  • d) \(-1{,}5\leq -\frac {300}{b}\leq -0{,}25\). Start med at argumentere for at \(b>0\).

Facit 1.12.6

  • a) \(2<x<6\)

  • b) \(1\leq x<2\)

  • c) \(100\leq a\leq 600\)

  • d) \(b>0\) fordi \(-\frac {300}{b}\) skal være negativ for at ulighed bliver sand. Løsningen til uligheden er: \(200\leq b \leq 1200\).

Øvelse 1.12.7

Vi vender tilbage til uligheden fra spørgsmål d) i ovenstående øvelse.

  • a) Hvorfor er det nødvendigt at argumentere for at \(b\) er positiv?

Facit 1.12.7

  • a) På et tidspunkt får du brug for at gange med \(b\) på begge sider. Hvis \(b<0\), ville du være nødt til at vende ulighedstegnet.

Øvelse 1.12.8

Opskriv løsningsmængden for dobbeltulighederne fra 1.12.6.

  • a) \(1<x-1<5\)

  • b) \(x+1\leq 2x<x+2\)

  • c) \(-1{,}5\leq -\frac {a}{400}\leq -0{,}25\)

  • d) \(-1{,}5\leq -\frac {300}{b}\leq -0{,}25\) (forudsæt at \(b>0\)).

Facit 1.12.8

  • a) \(]2;6[\)

  • b) \([1;2[\)

  • c) \([100;600]\)

  • d) \([200;1200]\)