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 \(2x+4<6x+2(x-4)\).
\(\seteqnumber{0}{1.}{0}\)
\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).
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 \(-5{,}5\)”.
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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)
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 \(4x\geq 2x+10\).
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:
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a) \(1<x-1<5\)
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b) \(x+1\leq 2x<x+2\)
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c) \(-1{,}5\leq -\frac {a}{400}\leq -0{,}25\)
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d) \(-1{,}5\leq -\frac {300}{b}\leq -0{,}25\). Start med at argumentere for at \(b>0\).
Øvelse 1.12.7
Vi vender tilbage til uligheden fra spørgsmål d) i ovenstående øvelse.
Øvelse 1.12.8
Opskriv løsningsmængden for dobbeltulighederne fra 1.12.6.
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a) \(1<x-1<5\)
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b) \(x+1\leq 2x<x+2\)
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c) \(-1{,}5\leq -\frac {a}{400}\leq -0{,}25\)
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d) \(-1{,}5\leq -\frac {300}{b}\leq -0{,}25\) (forudsæt at \(b>0\)).
Facit 1.12.8
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a) \(]2;6[\)
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b) \([1;2[\)
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c) \([100;600]\)
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d) \([200;1200]\)