Solving Equations and Inequalities - Revision history

Khanh at 04:17, 14 November 2021

2021-11-14T04:17:52Z

Lila: /* Licensing */

2021-10-21T17:33:13Z

Licensing

Lila: /* Resources and Examples */

2021-10-21T17:31:08Z

Resources and Examples

Lila: /* Resources and Examples */

2021-10-18T16:18:48Z

Resources and Examples

Lila: /* Neolithic Algebra */

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Neolithic Algebra

Lila: /* Manipulation of Radicals */

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Manipulation of Radicals

Lila: /* Inequalities */

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Inequalities

Lila: /* Introduction */

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Introduction

Lila: /* Inequalities */

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Inequalities

Show changes

Lila: /* Lesson Quiz */

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Lesson Quiz

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 * <math> -2x > 24 \implies (-2x)/(-2) < 24/(-2) \implies x < -12 </math>
-== Licensing ==
+= Licensing =
 Content obtained and/or adapted from:
 * [https://en.wikibooks.org/wiki/Algebra/Solving_Equations Solving Equations, Wikibooks: Algebra] under a CC BY-SA license
 * [https://en.wikibooks.org/wiki/Intermediate_Algebra/Solving_Inequalities Solving Inequalities, Wikibooks: Intermediate Algebra] under a CC BY-SA license

← Older revision		Revision as of 17:33, 21 October 2021
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	Content obtained and/or adapted from:		Content obtained and/or adapted from:
	* [https://en.wikibooks.org/wiki/Algebra/Solving_Equations Solving Equations, Wikibooks: Algebra] under a CC BY-SA license		* [https://en.wikibooks.org/wiki/Algebra/Solving_Equations Solving Equations, Wikibooks: Algebra] under a CC BY-SA license
		+	* [https://en.wikibooks.org/wiki/Intermediate_Algebra/Solving_Inequalities Solving Inequalities, Wikibooks: Intermediate Algebra] under a CC BY-SA license

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 ==Resources and Examples==
-See also: [Linear Equations]
+See also: [[Linear Equations]]
 ===General Resources===
 * [https://tutorial.math.lamar.edu/classes/alg/Solving.aspx Solving Equations and Inequalities], Paul's Online Notes
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 * <math> 6 - y \ge x \implies y - 6 \le -x \implies y \le  6 - x</math>
 * <math> -2x > 24 \implies (-2x)/(-2) < 24/(-2) \implies x < -12 </math>

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 ==Resources and Examples==
 ===General Resources===
 * [https://tutorial.math.lamar.edu/classes/alg/Solving.aspx Solving Equations and Inequalities], Paul's Online Notes

← Older revision		Revision as of 16:38, 4 October 2021
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	* Division - It is said that two things are inevitable: death and taxes. Division is the operation we use to make the inevitable fair. When a shepherd died his heirs could use division to divide up his flock fairly. They might use a function that allocated one sheep per heir until the flock was distributed. In what ways could the heirs distribute extra sheep if the flock was not evenly divisible? Similarly, when collecting taxes it is more fair to take a portion of the flock rather than a single amount. The clan leader would be better served taking 1/12th of a flock from each shepherd every year. If the clan leader asked for a specific number of sheep every year they might find that their clan became smaller every year just before tax time as new shepherds took their sheep to a new clan.		* Division - It is said that two things are inevitable: death and taxes. Division is the operation we use to make the inevitable fair. When a shepherd died his heirs could use division to divide up his flock fairly. They might use a function that allocated one sheep per heir until the flock was distributed. In what ways could the heirs distribute extra sheep if the flock was not evenly divisible? Similarly, when collecting taxes it is more fair to take a portion of the flock rather than a single amount. The clan leader would be better served taking 1/12th of a flock from each shepherd every year. If the clan leader asked for a specific number of sheep every year they might find that their clan became smaller every year just before tax time as new shepherds took their sheep to a new clan.
−
−	~~{{TextBox\|1= '''Thought Problem'''~~
−
−	Video Games have an attribute called [[w:game play\|game play]]. Look at a video game you enjoy playing and try to define how you interact with the environment using addition, subtraction, multiplication, and division.
−	}}

	== Using A Variable ==		== Using A Variable ==

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 == Manipulation of Radicals ==
-Let's say that we have a number <math>x</math>.  The square root of <math>x</math> is the number that, if multiplied by itself, equals <math>x</math>.   Since there are two numbers which satisfy that condition, we usually specify the positive value.  For example, the square root of 4 could be 2 (because <math>2 \times 2 = 4</math>) or it could be -2 (because <math>-2 \times -2 = 4</math>).  We use the symbol <math>\sqrt{x}</math> to indicate the [[Algebra/Square Root is Positive|positive]] square root of <math>x</math>.
+Let's say that we have a number <math>x</math>.  The square root of <math>x</math> is the number that, if multiplied by itself, equals <math>x</math>.   Since there are two numbers which satisfy that condition, we usually specify the positive value.  For example, the square root of 4 could be 2 (because <math>2 \times 2 = 4</math>) or it could be -2 (because <math>-2 \times -2 = 4</math>).  We use the symbol <math>\sqrt{x}</math> to indicate the positive square root of <math>x</math>.
 The cube root of <math>x</math> is the number that, if multiplied by itself three times, equals <math>x</math>.  We use the symbol <math>\sqrt[3]{x}</math> to indicate the cube root of <math>x</math>.

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 </quiz>
-== Inequalities ==
+= Introduction to Inequalities =
 As opposed to an equation, an '''inequality''' is an expression that states that two quantities are unequal or not equivalent to one another. In most cases we use inequalities in real life more than equations (i.e. this shirt costs $2 ''more than'' that one).

← Older revision		Revision as of 16:30, 4 October 2021
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−	== Introduction ==	+	= Introduction to Equations =

	We've seen that an equation is like a balance. When two quantities are on either side of an equal sign in a mathematical statement, we are saying that the statement is only true under conditions that make the quantities the same. For instance, when we see the statement <math>x+2=3</math> we now know that we are asking what number can we substitute for <math>x</math> in the equation to make this statement true? One way you could work this out is by trying out different values of <math>x</math> until you get one that works. This is called guess-and-check. Alternatively you might know the answer intuitively (by thinking ''What do I need to add to 2 to get 3?'').		We've seen that an equation is like a balance. When two quantities are on either side of an equal sign in a mathematical statement, we are saying that the statement is only true under conditions that make the quantities the same. For instance, when we see the statement <math>x+2=3</math> we now know that we are asking what number can we substitute for <math>x</math> in the equation to make this statement true? One way you could work this out is by trying out different values of <math>x</math> until you get one that works. This is called guess-and-check. Alternatively you might know the answer intuitively (by thinking ''What do I need to add to 2 to get 3?'').

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 == Lesson Review ==
 An inequality is a statement justifying that two quantities are not equal to each other. There are four cases of inequalities, two of which allow for equality (<math>a \le b</math> and <math>a \ge b</math>). The four properties of inequality, which are more or less parallel to the properties of equality, can be used to solve simple inequalities. The only exceptions are in multiplication and division, where the signs must be reversed if both sides are multiplied or divided by a negative number. Finally, the solution set of an inequality can be graphed on the number line with either an open circle ("O") or a closed circle ("*"), depending on the original sign used.
 ==Resources and Examples==