Solution 2.1:6b - Revision history

Tek at 11:35, 23 September 2008

2008-09-23T11:35:20Z

←Older revision		Revision as of 11:35, 23 September 2008
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-	The lowest common denominator for the three terms is	+	The lowest common denominator for the three terms is <math>(x-2)(x+3)</math> and we expand each term so that all terms have the same denominator
-	<math>~~\left~~( x-2 ~~\right~~)~~\left~~( x+3 ~~\right~~)</math>	+
-	and we expand each term so that all terms have the same denominator:	+

		+	{{Displayed math\|\|<math>\begin{align}
		+	\frac{x}{x-2}+\frac{x}{x+3}-2
		+	&= \frac{x}{x-2}\cdot\frac{x+3}{x+3} + \frac{x}{x+3}\cdot\frac{x-2}{x-2} - 2\cdot\frac{(x-2)(x+3)}{(x-2)(x+3)}\\[5pt]
		+	&= \frac{x(x+3)+x(x-2)-2(x-2)(x+3)}{(x-2)(x+3)}\\[5pt]
		+	&= \frac{x^{2}+3x+x^{2}-2x-2(x^{2}+3x-2x-6)}{(x-2)(x+3)}\\[5pt]
		+	&= \frac{x^{2}+3x+x^{2}-2x-2x^{2}-6x+4x+12}{(x-2)(x+3)}\,\textrm{.}
		+	\end{align}</math>}}

-	~~<math>\begin{align}~~	+	Now, collect the terms in the numerator
-	& \frac{x}{x-2}+\frac{x}{x+3}-2=\frac{x}{x-2}\centerdot \frac{x+3}{x+3}+\frac{x}{x+3}\centerdot \frac{x-2}{x-2}-2\centerdot \frac{\left( x-2 \right)\left( x+3 \right)}{\left( x-2 \right)\left( x+3 \right)} \\	+
-	~~& =\frac{x\left( x+3 \right)+x\left( x-2 \right)-2\left( x-2 \right)\left( x+3 \right)}{\left( x-2 \right)\left( x+3 \right)} \\~~	+
-	~~& =\frac{x^{2}+3x+x^{2}-2x-2\left( x^{2}+3x-2x-6 \right)}{\left( x-2 \right)\left( x+3 \right)} \\~~	+
-	~~& =\frac{x^{2}+3x+x^{2}-2x-2x^{2}-6x+4x+12}{\left( x-2 \right)\left( x+3 \right)} \\~~	+
-	~~\end{align}</math>~~	+

		+	{{Displayed math\|\|<math>\begin{align}
		+	\frac{x}{x-2}+\frac{x}{x+3}-2 &= \frac{(x^{2}+x^{2}-2x^{2})+(3x-2x-6x+4x)+12}{(x-2)(x+3)}\\[5pt]
		+	&= \frac{-x+12}{(x-2)(x+3)}\,\textrm{.}
		+	\end{align}</math>}}

-	~~Now, collect together the terms in the numerator:~~	+	Note: By keeping the denominator factorized during the entire calculation, we can see at the end that the answer cannot be simplified any further.
-		+
-		+
-	~~<math>\begin{align}~~	+
-	~~& \frac{x}{x-2}+\frac{x}{x+3}-2=\frac{\left( x^{2}+x^{2}-2x^{2} \right)+\left( 3x-2x-6x+4x \right)+12}{\left( x-2 \right)\left( x+3 \right)} \\~~	+
-	~~& =\frac{-x+12}{\left( x-2 \right)\left( x+3 \right)} \\~~	+
-	~~\end{align}</math>~~	+
-		+
-		+
-	~~NOTE~~: By keeping the denominator factorized during the entire calculation, we can see at the end that the answer cannot be simplified any further.	+

Ian at 10:53, 16 September 2008

2008-09-16T10:53:42Z

←Older revision		Revision as of 10:53, 16 September 2008
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-	{{~~NAVCONTENT_START~~}}	+	The lowest common denominator for the three terms is
-	<~~center~~> ~~[[Image~~:~~2_1_6b.gif]]~~ <~~/center~~>	+	<math>\left( x-2 \right)\left( x+3 \right)</math>
-	{{~~NAVCONTENT_STOP~~}}	+	and we expand each term so that all terms have the same denominator:
		+
		+
		+	<math>\begin{align}
		+	& \frac{x}{x-2}+\frac{x}{x+3}-2=\frac{x}{x-2}\centerdot \frac{x+3}{x+3}+\frac{x}{x+3}\centerdot \frac{x-2}{x-2}-2\centerdot \frac{\left( x-2 \right)\left( x+3 \right)}{\left( x-2 \right)\left( x+3 \right)} \\
		+	& =\frac{x\left( x+3 \right)+x\left( x-2 \right)-2\left( x-2 \right)\left( x+3 \right)}{\left( x-2 \right)\left( x+3 \right)} \\
		+	& =\frac{x^{2}+3x+x^{2}-2x-2\left( x^{2}+3x-2x-6 \right)}{\left( x-2 \right)\left( x+3 \right)} \\
		+	& =\frac{x^{2}+3x+x^{2}-2x-2x^{2}-6x+4x+12}{\left( x-2 \right)\left( x+3 \right)} \\
		+	\end{align}</math>
		+
		+
		+	Now, collect together the terms in the numerator:
		+
		+
		+	<math>\begin{align}
		+	& \frac{x}{x-2}+\frac{x}{x+3}-2=\frac{\left( x^{2}+x^{2}-2x^{2} \right)+\left( 3x-2x-6x+4x \right)+12}{\left( x-2 \right)\left( x+3 \right)} \\
		+	& =\frac{-x+12}{\left( x-2 \right)\left( x+3 \right)} \\
		+	\end{align}</math>
		+
		+
		+	NOTE: By keeping the denominator factorized during the entire calculation, we can see at the end that the answer cannot be simplified any further.

Tekbot: Lösning 2.1:6b moved to Solution 2.1:6b: Robot: moved page

2008-09-09T08:39:23Z

Lösning 2.1:6b moved to Solution 2.1:6b: Robot: moved page

←Older revision

Revision as of 08:39, 9 September 2008

Tekbot: Robot: Automated text replacement (-[[Bild: +[[Image:)

2008-08-21T06:36:49Z

Robot: Automated text replacement (-[[Bild: +[[Image:)

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2008-03-31T11:16:13Z

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