Sammanställning övningar

Sommarmatte 1

(Skillnad mellan versioner)

Versionen från 21 juni 2007 kl. 15.07

Övning 1.1:1

Beräkna (utan hjälp av räknedosa)

a)	$3-7-4+6-5$	b)	$3-(7-4)+(6-5)$
c)	$3-(7-(4+6)-5)$	d)	$3-(7-(4+6))-5$

a)	$-7$	b)	$1$
c)	$11$	d)	$1$

Övning 1.1:2

Beräkna (utan hjälp av räknedosa)

a)	$(3-(7-4))(6-5)$	b)	$3-(((7-4)+6)-5)$
c)	$3\cdot(-7)-4\cdot(6-5)$	d)	$3\cdot(-7)-(4+6)/(-5)$

a)	$0$	b)	$-1$
c)	$-25$	d)	$-19$

Övning 1.1:3

Vilka av följande tal tillhör de naturliga talen? heltalen? rationella talen? irrationella talen? Förenkla först!

a)	$8$	b)	$-4$	c)	$8-4$
d)	$4-8$	e)	$8(-4)$	f)	$(-8)(-4)$
g)	$\displaystyle \frac{4}{-8}$	h)	$\displaystyle \frac{-8}{-4}$	i)	$\displaystyle \frac{\sqrt{2}}{3}$
j)	$\displaystyle \Bigl(\frac{4}{\sqrt{2}}\Bigr)^2$	k)	$-\pi$	l)	$\pi+1$


a)	naturliga talen, heltalen, rationella talen	b)	heltalen, rationella talen	c)	naturliga talen, heltalen, rationella talen
d)	heltalen, rationella talen	e)	heltalen, rationella talen	f)	naturliga talen, heltalen, rationella talen
g)	rationella talen	h)	naturliga talen, heltalen, rationella talen	i)	irrationella talen
j)	naturliga talen, heltalen, rationella talen	k)	irrationella talen	l)	irrationella talen

Övning 1.1:4

Ordna följande tal i storleksordning

a)	$\displaystyle 2,\ \frac{3}{5},\ \frac{5}{3}\ $ och $\ \displaystyle \frac{7}{3}$
b)	$\displaystyle -\frac{1}{2},\ -\frac{1}{5},\ -\frac{3}{10}\ $ och $\ \displaystyle -\frac{1}{3}$
c)	$\displaystyle \frac{1}{2},\ \frac{2}{3},\ \frac{3}{5},\ \frac{5}{8}\ $ och $\ \displaystyle \frac{21}{34}$

a)	$\displaystyle \frac{3}{5}<\frac{5}{3}<2<\frac{7}{3}$
b)	$\displaystyle -\frac{1}{2}<-\frac{1}{3}<-\frac{3}{10}<-\frac{1}{5}$
c)	$\displaystyle \frac{1}{2}<\frac{3}{5}<\frac{21}{34}<\frac{5}{8}<\frac{2}{3}$

Övning 1.1:5

Ange decimalutvecklingen med tre korrekta decimaler till

a)	$\displaystyle \frac{7}{6}$	b)	$\displaystyle \frac{9}{4}$	c)	$\displaystyle \frac{2}{7}$	d)	$\sqrt{2}$

a)	$1{,}167$	b)	$2{,}250$	c)	$0{,}286$	d)	$1{,}414$

Övning 1.1:6

Vilka av följande tal är rationella? Ange dem som en kvot mellan heltal.

a)	$3,14$
b)	$3{,}1416\,1416\,1416\,\dots$
c)	$0{,}2\,001\,001\,001\,\dots\,$ (därefter är var tredje decimal en 1:a och övriga 0)
d)	$0{,}10\,100\,1000\,10000\,1\dots\, $ (en 1:a, en 0:a, en 1:a, två 0:or, en 1:a, tre 0:or osv.)

a)	Talet är rationellt och lika med $\,314/100 = 157/50\,$.
b)	Talet är rationellt och är lika med $\,31413/9999 = 10471/3333\,$.
c)	Talet är rationellt och lika med $\,1999/9990\,$.
d)	Talet är irrationellt.

Övning 1.2:1

Skriv på gemensamt bråkstreck

a)	$\displaystyle \frac{7}{4}+\frac{11}{7}$	b)	$\displaystyle \frac{2}{7}-\frac{1}{5}$	c)	$\displaystyle \frac{1}{6}-\frac{2}{5}$
d)	$\displaystyle \frac{1}{3}+\frac{1}{4}+\frac{1}{5}$	e)	$\displaystyle \frac{8}{7}+\frac{3}{4}-\frac{4}{3}$

a)	$\displaystyle \frac{93}{28}$	b)	$\displaystyle \frac{3}{35}$	c)	$\displaystyle -\frac{7}{30}$
d)	$\displaystyle \frac{47}{60}$	e)	$\displaystyle \frac{47}{84}$

Övning 1.2:2

Bestäm minsta gemensamma nämnare

a)	$\displaystyle \frac{1}{6}+\frac{1}{10}$	b)	$\displaystyle \frac{1}{4}-\frac{1}{8}$
c)	$\displaystyle \frac{1}{12}-\frac{1}{14}$	d)	$\displaystyle \frac{2}{45}+\frac{1}{75}$

a)	$\displaystyle {30}$	b)	$\displaystyle {8}$
c)	$\displaystyle {84}$	d)	$\displaystyle {225}$

Övning 1.2:3

Beräkna följande uttryck genom att använda minsta gemensamma nämnare:

a)	$\displaystyle \frac{3}{20}+\frac{7}{50}-\frac{1}{10}$	b)	$\displaystyle \frac{1}{24}+\frac{1}{40}-\frac{1}{16}$


a)	$\displaystyle \frac{19}{100}$	b)	$\displaystyle \frac{1}{240}$

Övning 1.2:4

Förenkla följande uttryck genom att skriva på gemensamt bråkstreck. Bråket ska vara färdigförkortat.

a)	$\displaystyle\frac{\displaystyle\frac{3}{5}}{\displaystyle\frac{7}{10}}$	b)	$\displaystyle\frac{\displaystyle\frac{2}{7}}{\displaystyle\frac{3}{8}}$	c)	$\displaystyle\frac{\displaystyle\frac{1}{4}-\frac{1}{5}}{\displaystyle\frac{3}{10}}$

a)	$\displaystyle \frac{6}{7}$	b)	$\displaystyle \frac{16}{21}$	c)	$\displaystyle \frac{1}{6}$

Övning 1.2:5

Förenkla följande uttryck genom att skriva på gemensamt bråkstreck. Bråket ska vara färdigförkortat.

a)	$\displaystyle \frac{2}{\displaystyle \frac{1}{7}\displaystyle -\frac{1}{15}}$	b)	$\displaystyle\frac{\displaystyle\frac{1}{2}\displaystyle+\frac{1}{3}}{\displaystyle\frac{1}{3}\displaystyle-\frac{1}{2}}$	c)	$\displaystyle\frac{\displaystyle\frac{3}{10}\displaystyle-\frac{1}{5}}{\displaystyle\frac{7}{8}\displaystyle-\frac{3}{16}}$


a)	$\displaystyle \frac{105}{4}$	b)	$-5$	c)	$\displaystyle \frac{8}{55}$

Övning 1.2:6

Förenkla $\ \,\displaystyle \frac{\displaystyle \frac{2}{\displaystyle 3+\frac{1}{2}}\displaystyle + \frac{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{4}\displaystyle -\frac{1}{3}}}{\displaystyle \frac{1}{2}\displaystyle - \frac{3}{\displaystyle 2-\frac{2}{7}}}$

$\displaystyle \frac{152}{35}$

Övning 1.3:1

Beräkna

a)	$2^3\cdot3^2$	b)	$3^5\cdot9^{-2}$	c)	$(-5)^3$	d)	$\Bigl(\displaystyle \frac{2}{3}\Bigr)^{-3}$

a)	$72$	b)	$3$	c)	$-125$	d)	$\displaystyle \frac{27}{8}$

Övning 1.3:2

Skriv som en potens av 2

a)	$2\cdot4\cdot8$	b)	$0{,}25$	c)	$1$

a)	$2^6$	b)	$2^{-2}$	c)	$2^0$

Övning 1.3:3

Skriv som en potens av 3

a)	$\displaystyle \frac{1}{3}$	b)	$243$	c)	$9^2$	d)	$\displaystyle \frac{1}{27}$	e)	$\displaystyle \frac{3}{9^2}$

a)	$3^{-1}$	b)	$3^5$	c)	$3^4$	d)	$3^{-3}$	e)	$3^{-3}$

Övning 1.3:4

Beräkna

a)	$2^{9}\cdot2^{-7}$	b)	$3^{13}\cdot9^{-3}\cdot27^{\,-2}$	c)	$\displaystyle \frac{5^{12}}{5^{-4}}\cdot(5^{2})^{-6}$
d)	$2^{2^{\scriptstyle3}}\cdot(-2)^{\scriptstyle-4}$	e)	$625\cdot(5^{8}+5^{9})^{-1}$

a)	$4$	b)	$3$	c)	$625$
d)	$16$	e)	$\displaystyle \frac{1}{3750}$

Övning 1.3:5

Beräkna

a)	$4^{1/2}$	b)	$4^{-1/2}$	c)	$9^{3/2}$
d)	$\left(47^{2/3} \right) ^{3}$	e)	$3^{1{,}4}\cdot3^{0{,}6}$	f)	$\bigl( 125 ^{1/3} \bigr)^2\cdot \bigl( 27^{1/3} \bigr)^{-2}\cdot9^{1/2}$

a)	$2$	b)	$\displaystyle \frac{1}{2}$	c)	$27$
d)	$2209$	e)	$9$	f)	$\displaystyle \frac{25}{3}$

Övning 1.3:6

Avgör vilket tal som är störst av

a)	$256^{1/3}\ $ och $\ 200^{1/3}$	b)	$0{,}5^{-3}\ $ och $\ 0{,}4^{-3}$	c)	$0{,}2^5\ $ och $\ 0{,}2^{7}$
d)	$400^{1/3}\ $ och $\ \bigl(5^{1/3}\bigr)^{4}$	e)	$125^{1/2}\ $ och $\ 625^{1/3}$	f)	$2^{56}\ $ och $\ 3^{40}$

a)	$256^{1/3}>200^{1/3}$	b)	$0{,}4^{-3}>0{,}5^{-3}$	c)	$0{,}2^{5}>0{,}2^{7}$
d)	$\bigl(5^{1/3}\bigr)^{4}>400^{1/3}$	e)	$125^{1/2}>625^{1/3}$	f)	$3^{40}>2^{56}$

Den här artikeln är hämtad från http://wiki.math.se/wikis/sf0600_0701/index.php/Sammanst%C3%A4llning_%C3%B6vningar

Sammanställning övningar

Sommarmatte 1

Versionen från 21 juni 2007 kl. 15.07

Övning 1.1:1

Övning 1.1:2

Övning 1.1:3

Övning 1.1:4

Övning 1.1:5

Övning 1.1:6

Övning 1.2:1

Övning 1.2:2

Övning 1.2:3

Övning 1.2:4

Övning 1.2:5

Övning 1.2:6

Övning 1.3:1

Övning 1.3:2

Övning 1.3:3

Övning 1.3:4

Övning 1.3:5

Övning 1.3:6

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 __NOTOC__
-[[övn 2.3]]
+[[övn 2]]
 ==Övning 1.1:1==
 <td class="ntext">f)</td>
 <td class="ntext" width="33%">$3^{40}>2^{56}$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-==Övning 2.1:1==
-<div class="ovning">
-Utveckla
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">$3x(x-1)$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">$(1+x-x^2)xy$</td>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">$-x^2(4-y^2)$</td>
-</tr>
-<tr align="left">
-<td class="ntext">d)</td>
-<td class="ntext" width="33%">$x^3y^2\left(\displaystyle \frac{1}{y} - \frac{1}{xy}+1\right)$</td>
-<td class="ntext">e) </td>
-<td class="ntext" width="33%">$(x-7)^2$</td>
-<td class="ntext">f)</td>
-<td class="ntext" width="33%">$(5+4y)^2$</td>
-</tr>
-<tr align="left">
-<td class="ntext">g)</td>
-<td class="ntext" width="33%">$(y^2-3x^3)^2$</td>
-<td class="ntext">h)</td>
-<td class="ntext" width="33%">$(5x^3+3x^5)^2$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">$3x^2-3x$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">$xy+x^2y-x^3y$</td>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">$-4x^2+x^2y^2$</td>
-</tr>
-<tr align="left">
-<td class="ntext">d)</td>
-<td class="ntext" width="33%">$x^3y-x^2y+x^3y^2$</td>
-<td class="ntext">e)</td>
-<td class="ntext" width="33%">$x^2-14x+49$</td>
-<td class="ntext">f)</td>
-<td class="ntext" width="33%">$16y^2+40y+25$</td>
-</tr>
-<tr align="left">
-<td class="ntext">g)</td>
-<td class="ntext" width="33%">$9x^6-6x^3y^2+y^4$</td>
-<td class="ntext">h)</td>
-<td class="ntext" width="33%">$9x^{10}+30x^8+25x^6$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-==Övning 2.1:2==
-<div class="ovning">
-Utveckla och förenkla så långt som möjligt
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$(x-4)(x-5)-3x(2x-3)$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$(1-5x)(1+15x)-3(2-5x)(2+5x)$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="50%">$(3x+4)^2-(3x-2)(3x-8)$</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$(3x^2+2)(3x^2-2)(9x^4+4)$</td>
-</tr>
-<tr align="left">
-<td class="ntext">e)</td>
-<td class="ntext" width="50%">$(a+b)^2+(a-b)^2$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$-5x^2+20$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$10x-11$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="50%">$54x$</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$81x^8-16$</td>
-</tr>
-<tr align="left">
-<td class="ntext">e)</td>
-<td class="ntext" width="50%">$2a^2+2b^2$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-==Övning 2.1:3==
-<div class="ovning">
-Faktorisera s&aring; l&aring;ngt som m&ouml;jligt
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">$x^2-36$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">$5x^2-20$</td>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">$x^2+6x+9$</td>
-</tr>
-<tr align="left">
-<td class="ntext">d)</td>
-<td class="ntext" width="33%">$x^2-10x+25$</td>
-<td class="ntext">e)</td>
-<td class="ntext" width="33%">$18x-2x^3$</td>
-<td class="ntext">f)</td>
-<td class="ntext" width="33%">$16x^2+8x+1$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">$(x+6)(x-6)$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">$5(x+2)(x-2)$</td>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">$(x+3)^2$</td>
-</tr>
-<tr align="left">
-<td class="ntext">d)</td>
-<td class="ntext" width="33%">$(x-5)^2$</td>
-<td class="ntext">e)</td>
-<td class="ntext" width="33%">$-2x(x+3)(x-3)$</td>
-<td class="ntext">f)</td>
-<td class="ntext" width="33%">$(4x+1)^2$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-==Övning 2.1:4==
-<div class="ovning">
-Bestäm koefficienterna framför $\,x\,$ och $\,x^2\,$ när följande uttryck utvecklas
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="100%">$(x+2)(3x^2-x+5)$</td>
-</tr>
-<tr align="left">
-<td class="ntext">b)</td>
-<td class="ntext" width="100%">$(1+x+x^2+x^3)(2-x+x^2+x^4)$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="100%">$(x-x^3+x^5)(1+3x+5x^2)(2-7x^2-x^4)$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="100%">$5\,$ framf&ouml;r $\,x^2\,$, $\,3\,$ framf&ouml;r $\,x$</td>
-</tr>
-<tr align="left">
-<td class="ntext">b)</td>
-<td class="ntext" width="100%">$2\,$ framf&ouml;r $\,x^2\,$, $\,1\,$ framf&ouml;r $\,x$</td>
-</tr>
-<tr align="left">
-<td class="ntext">$\textrm{c) }$</td>
-<td class="ntext" width="100%">$6\,$ framf&ouml;r $\,x^2\,$, $\,2\,$ framf&ouml;r $\,x$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-==Övning 2.1:5==
-<div class="ovning">
-Förenkla så långt som möjligt
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$\displaystyle \frac{1}{x-x^2}-\displaystyle \frac{1}{x}$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$\displaystyle \frac{1}{y^2-2y}-\displaystyle \frac{2}{y^2-4}$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="50%">$\displaystyle \frac{(3x^2-12)(x^2-1)}{(x+1)(x+2)}$</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$\displaystyle \frac{(y^2+4y+4)(2y-4)}{(y^2+4)(y^2-4)}$</td>
-</tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$\displaystyle \frac{1}{1-x}$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$-\displaystyle \frac{1}{y(y+2)}$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="50%">$3(x-2)(x-1)$</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$\displaystyle \frac{2(y+2)}{y^2+4}$</td>
-</tr>
-</table>
-</div>
-==Övning 2.1:6==
-<div class="ovning">
-Förenkla så långt som möjligt
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$\left(x-y+\displaystyle\frac{x^2}{y-x}\right)$ $\left(\displaystyle\frac{y}{2x-y}-1\right)$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$\displaystyle \frac{x}{x-2}+\displaystyle \frac{x}{x+3}-2$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="50%">$\displaystyle \frac{2a+b}{a^2-ab}-\frac{2}{a-b}$</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$\displaystyle\frac{a-b+\displaystyle\frac{b^2}{a+b}}{1-\left(\displaystyle\frac{a-b}{a+b}\right)^2}$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$2y$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$\displaystyle\frac{-x+12}{(x-2)(x+3)}$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="50%">$\displaystyle\frac{b}{a(a-b)}$</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$\displaystyle\frac{a(a+b)}{4b}$</td>
-</tr>
-</table>
-</div>
-==Övning 2.1:7==
-<div class="ovning">
-Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">$\displaystyle \frac{2}{x+3}-\frac{2}{x+5}$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">$x+\displaystyle \frac{1}{x-1}+\displaystyle \frac{1}{x^2}$</td>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">$\displaystyle \frac{ax}{a+1}-\displaystyle \frac{ax^2}{(a+1)^2}$</td>
-</tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">$\displaystyle \frac{4}{(x+3)(x+5)}$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">$\displaystyle \frac{x^4-x^3+x^2+x-1}{x^2(x-1)}$</td>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">$\displaystyle \frac{ax(a+1-x)}{(a+1)^2}$</td>
-</tr>
-</table>
-</div>
-==Övning 2.1:8==
-<div class="ovning">
-Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck
-<table width="100%" cellspacing="10px">
-<tr>
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">$\displaystyle \frac{\displaystyle\ \frac{x}{x+1}\ }{\ 3+x\ }$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">$\displaystyle \frac{\displaystyle \frac{3}{x}-\displaystyle \frac{1}{x}}{\displaystyle \frac{1}{x-3}}$</td>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">$\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+x}}}$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">$\displaystyle \frac{x}{(x+3)(x+1)}$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">$\displaystyle \frac{2(x-3)}{x}$</td>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">$\displaystyle \frac{x+2}{2x+3}$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-==Övning 2.2:1==
-<div class="ovning">
-L&ouml;s ekvationerna
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$x-2=-1$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$2x+1=13$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="50%">$\displaystyle\frac{1}{3}x-1=x$</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$5x+7=2x-6$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$x=1$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$x=6$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c) </td>
-<td class="ntext" width="50%">$x=-\displaystyle\frac{3}{2}$</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$x=-\displaystyle\frac{13}{3}$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-==Övning 2.2:2==
-<div class="ovning">
-L&ouml;s ekvationerna
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$\displaystyle\frac{5x}{6}-\displaystyle\frac{x+2}{9}=\displaystyle\frac{1}{2}$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$\displaystyle\frac{8x+3}{7}-\displaystyle\frac{5x-7}{4}=2$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="50%">$(x+3)^2-(x-5)^2=6x+4$</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$(x^2+4x+1)^2+3x^4-2x^2=(2x^2+2x+3)^2$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$x=1$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$x=\displaystyle\frac{5}{3}$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="50%">$x=2$</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$x=-2$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-==Övning 2.2:3==
-<div class="ovning">
-L&ouml;s ekvationerna
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="100%">$\displaystyle\frac{x+3}{x-3}-\displaystyle\frac{x+5}{x-2}=0$</td>
-</tr>
-<tr align="left">
-<td class="ntext">b)</td>
-<td class="ntext" width="100%">$\displaystyle\frac{4x}{4x-7}-\displaystyle\frac{1}{2x-3}=1$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="100%">$\left(\displaystyle\frac{1}{x-1}-\frac{1}{x+1}\right)\left(x^2+\frac{1}{2}\right)=\displaystyle\frac{6x-1}{3x-3}$</td>
-</tr>
-<tr align="left">
-<td class="ntext">d) </td>
-<td class="ntext" width="100%"> $\left(\displaystyle\frac{2}{x}-3\right)\left(\displaystyle\frac{1}{4x}+\frac{1}{2}\right)-\left(\displaystyle\frac{1}{2x}-\frac{2}{3}\right)^2-\left(\displaystyle\frac{1}{2x}+\frac{1}{3}\right)\left(\displaystyle\frac{1}{2x}-\frac{1}{3}\right)=0$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="100%">$x=9$</td>
-</tr>
-<tr align="left">
-<td class="ntext">b)</td>
-<td class="ntext" width="100%">$x=\displaystyle\frac{7}{5}$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c) </td>
-<td class="ntext" width="100%">$x=\displaystyle\frac{4}{5}$</td>
-</tr>
-<tr align="left">
-<td class="ntext">d) </td>
-<td class="ntext" width="100%">$x=\displaystyle\frac{1}{2}$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-==&Ouml;vning 2.2:4==
-<div class="ovning">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td>a)</td>
-<td width="100%">Skriv ekvationen f&ouml;r linjen $\,y=2x+3\,$ på formen $\,ax+by=c\,$.</td>
-</tr>
-<tr align="left">
-<td>b)</td>
-<td width="100%">Skriv ekvationen f&ouml;r linjen $\,3x+4y-5=0\,$ på formen $\,y=kx+m\,$.</td>
-</tr>
-<tr><td height="5px"\></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td>a)</td>
-<td width="100%">$-2x+y=3$</td>
-</tr>
-<tr align="left">
-<td>b)</td>
-<td width="100%">$y=-\displaystyle\frac{3}{4}x+\frac{5}{4}$</td>
-</tr>
-<tr><td height="5px"\></tr>
-</table>
-</div>
-==&Ouml;vning 2.2:5==
-<div class="ovning">
-<table width="100%" cellspacing="10px">
-<tr align="left" valign="top">
-<td class="ntext">a)</td>
-<td class="ntext" width="100%">Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som g&aring;r genom punkterna $\,(2,3)\,$ och $\,(3,0)\,$.</td>
-</tr>
-<tr align="left" valign="top">
-<td class="ntext">b)</td>
-<td class="ntext" width="100%">Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som har riktningskoefficient $\,-3\,$ och g&aring;r genom punkten $\,(1,-2)\,$.</td>
-</tr>
-<tr align="left" valign="top">
-<td class="ntext">c)</td>
-<td class="ntext" width="100%">Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som g&aring;r genom punkten $\,(-1,2)\,$ och &auml;r parallell med linjen $\,y=3x+1\,$.</td>
-</tr>
-<tr align="left" valign="top">
-<td class="ntext">d)</td>
-<td class="ntext" width="100%">Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som g&aring;r genom punkten $\,(2,4)\,$ och &auml;r vinkelr&auml;t mot linjen $\,y=2x+5\,$.</td>
-</tr>
-<tr align="left" valign="top">
-<td class="ntext">e)</td>
-<td class="ntext" width="100%">Best&auml;m riktningskoefficienten, $\,k\,$, f&ouml;r den r&auml;ta linje som sk&auml;r ''x''-axeln i punkten $\,(5,0)\,$ och ''y''-axeln i punkten $\,(0,-8)\,$.</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="100%">$y=-3x+9$</td>
-</tr>
-<tr align="left">
-<td class="ntext">b)</td>
-<td class="ntext" width="100%">$y=-3x+1$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="100%">$y=3x+5$</td>
-</tr>
-<tr align="left">
-<td class="ntext">d)</td>
-<td class="ntext" width="100%">$y=-\displaystyle \frac{1}{2}x+5$</td>
-</tr>
-<tr align="left">
-<td class="ntext">e)</td>
-<td class="ntext" width="100%">$k = \displaystyle\frac{8}{5}$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-==&Ouml;vning 2.2:6==
-<div class="ovning">
-Finn sk&auml;rningspunkten mellan f&ouml;ljande linjer
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$y=3x+5\ $ och ''x''-axeln</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$y=-x+5\ $ och ''y''-axeln</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="50%">$4x+5y+6=0\ $ och ''y''-axeln</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$x+y+1=0\ $ och $\ x=12$</td>
-</tr>
-<tr align="left">
-<td class="ntext">e)</td>
-<td class="ntext" width="50%">$2x+y-1=0\ $ och $\ y-2x-2=0$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="50%">$\bigl(-\frac{5}{3},0\bigr)$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="50%">$(0,5)$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="50%">$\bigl(0,-\frac{6}{5}\bigr)$</td>
-<td class="ntext">d)</td>
-<td class="ntext" width="50%">$(12,-13)$</td>
-</tr>
-<tr align="left">
-<td class="ntext">e)</td>
-<td class="ntext" width="50%">$\bigl(-\frac{1}{4},\frac{3}{2}\bigr)$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-==&Ouml;vning 2.2:7==
-<div class="ovning">
-Skissera grafen till f&ouml;ljande funktioner
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">$f(x)=3x-2$</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">$f(x)=2-x$</td>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">$f(x)=2$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_7a.gif‎]] </td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_7b.gif‎]] </td>
-</tr><tr>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_7c.gif‎]] </td>
-</tr>
-</table>
-</div>
-==&Ouml;vning 2.2:8==
-<div class="ovning">
-Rita in i ett ''xy''-plan alla punkter vars koordinater $\,(x,y)\,$ uppfyller
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">$y \geq x $</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">$y &lt; 3x -4 $</td>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">$2x+3y \leq 6 $</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_8a.gif‎]]</td>
-<td class="ntext">b)</td>
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_8b.gif‎]]</td>
-</tr><tr>
-<td class="ntext">c)</td>
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_8c.gif‎]]</td>
-</tr>
-</table>
-</div>
-==&Ouml;vning 2.2:9==
-<div class="ovning">
-Ber&auml;kna arean av den triangel som
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="100%">har h&ouml;rn i punkterna $\,(1,4)\,$, $\,(3,3)\,$ och $\,(1,0)\,$</td>
-</tr>
-<tr align="left">
-<td class="ntext">b)</td>
-<td class="ntext" width="100%">begr&auml;nsas av linjerna $\ x=2y\,$, $\ y=4\ $ och $\ y=10-2x\,$</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="100%">beskrivs av olikheterna $\ x+y \geq -2\,$, $\ 2x-y \leq 2\ $ och $\ 2y-x \leq 2\,$</td>
-</tr>
-<tr><td height="5px"/></tr>
-</table>
-</div>
-<div class="svar">
-<table width="100%" cellspacing="10px">
-<tr align="left">
-<td class="ntext">a)</td>
-<td class="ntext" width="100%">$4\,$ a.e.</td>
-</tr>
-<tr align="left">
-<td class="ntext">b)</td>
-<td class="ntext" width="100%">$5\,$ a.e.</td>
-</tr>
-<tr align="left">
-<td class="ntext">c)</td>
-<td class="ntext" width="100%">$6\,$ a.e.</td>
 </tr>
 <tr><td height="5px"/></tr>
 </table>
 </div>