text:
Algebra/Chapter 3/Solving Equations

Introduction
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 $$x+2=3$$ we now know that we are asking what number can we substitute for $$x$$ in the equation to make this statement true? One way you could work this out is by trying out different values of $$x$$ 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?).

However, if you have a more complicated problem such as $$\frac{7x}{2} + 100 = 170$$ you are likely to have trouble solving this problem intuitively or by guess-and-check. Because of this, mathematicians worked out a technique to solve this type of problem easily. This technique is the fundamental basis of algebra.

Since the equal sign means that both sides of the equation are the same (Same value, different appearance), and that if you manipulate (using addition, multiplication, etc.) the values on both sides of the equal sign in the same way, then they will still be equal. This is also how a mechanical balance works. As long as you do the same thing to whatever is in both pans they stay level. If we have a value of 3 = in one pan and two things with the value of 2 and 1 in the other pan the balance is level. If we multiply both sides by 5 we get,

 * $$3 \times 5 = (2 + 1) \times 5$$
 * $$15 = (2 \times 5) + (1 \times 5)$$
 * $$15 = 10 + 5$$
 * $$15 = 15$$

Notice how the equality still holds.

Neolithic Algebra
Neolithic Functions Imagine you are the neolithic shepherd. What changes can you make to your bag of rocks?

Possible Answers:
 * Addition - When new sheep are added to your flock you would use addition. For instance in the spring you would have a "lamb born" function in which you add a rock to your bag.  You might also have a "multiple birth" function in which you invoke the lamb born function once for each lamb born.

 * Subtraction - When you decide that a sheep is no longer part of your flock you would use a "remove sheep function" to keep your bag of rocks in balance with your herd. You might do different things with the rock for a missing sheep, a sheep you traded for something you needed, or because it became dinner one night. You might anticipate using these functions to manage your herd.  If you know that you need to remove five rocks from your bag in the fall to buy firewood you might examine your bag carefully to know how many times you can invoke the "sheep dinner" function over the summer.
 * Multiplication - Imagine you are a shepherd just starting out. If your neighbors had a big enough flock they might promise you a number of sheep per month as payment for watching their flock. For instance if they promised you two sheep a month for tending their flock then you would know that after 1 month you would have two sheep, after two months you would have four sheep, and after an entire year you might have enough sheep to start your own operation.

 * 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.

Using A Variable
With algebra we use variables to represent things we haven't been able to count. Since we can't count the number the variable represents the number is called an "unknown". The most common variable is denoted $$x$$. To use a variable we write an expression which we know is true and has $$x$$ on its own on one side of an equals sign and a number on the other side of the equals sign.

Sometimes a variable can represent a set of numbers. In this case the number can be represented with set notation. We generally use a letter that reminds us what the variable represents.

Here are some examples of manipulating equations to get the $$x$$ on its own,

Example 1: How many dollars do I need to see a movie and buy popcorn? If we let $$x$$ equal the dollars that I need to get into the theater, then we could use the variable set $$A={3,5,7,10}$$ to represent a discount admission, a matinee admission, a regular admission, or an Imax admission. If we assume that popcorn costs 3 dollars at all the theaters we will go to then we can write the equation $$x - A = 3$$ to represent the money we need. We could add an A to both sides to get the equation $$x = 3 + A$$. Plugging the values for A into this equation we find that $$x= {6, 8, 10, 13}$$.

Example 2: What is the value of $$x$$ in $$2x = 4$$?

Solution: We can do the same in this case by dividing it by 2 (as $$\frac{2x}{2}$$ is $$x$$ (remember that $$x$$ is the same as 1$$x$$ because it has implied coefficient of 1)), but again to keep both sides of the equation equal we'll need to divide the other side by 2 as well to get,

 * $$\frac{2x}{2} = \frac{4}{2}$$
 * $$x = \frac{4}{2}$$
 * $$x = 2$$

Example 3: What is the value of $$x$$ in $$3x + 1 = 4$$

Solution: Here we first need to subtract 1 from both sides,

 * $$3x + 1 - 1 = 4 - 1$$
 * $$3x = 3$$

Then we divide both sides by 3 to get,

 * $$\frac{3x}{3} = \frac{3}{3}$$
 * $$x = 1$$

Although in this case we chose to do the subtraction first and then the division, we could have done it the other way around, doing the division first followed by the subtraction, as follows,

 * $$\frac{3x}{3} + \frac{1}{3} = \frac{4}{3}$$
 * $$x + \frac{1}{3} = \frac{4}{3}$$
 * $$x = \frac{4}{3} - \frac{1}{3}$$
 * $$x = \frac{3}{3} $$
 * $$x = 1$$

When working problems it often helps to do additions/subtractions first and then multiplication/divisions, as this lets us avoid having to add or subtract fractions. However, both ways are equally valid.

Simplifying Equations
Sometimes you'll come across equations which have a variable on both sides, for instance $$5x-1=2x+2$$, where x can be found on both sides of the equation.

We solve this type of equation in much the same way as we've solved the previous problems, but only this time you have to first make sure all of the variables are on the same side. The easiest way to see how to do this is by example:

Example 1: How do you find the value of $$x$$ in the equation, $$5x-1=2x+2$$?

First of all you need to choose which side you want the variable to be on, the left or the right of the equals sign, in this case we'll choose to have the $$x$$ on the left hand side.

To do this we first have to look at where $$x$$ occurs on the right hand side; in this case it only appears in the term $$2x$$. As we don't want $$x$$ on the right hand side we need to get rid of it, and we can do this by subtracting $$2x$$ from the right side. Remember that for the equality to still be accurate we need to do the same on the left side as well.

 * $$5x-1-2x=2x+2-2x$$ (subtracting 2x from both sides)
 * $$3x-1=2$$ (simplifying)

Now the equation is in a form which you are familiar with from the last chapter so hopefully you should now be able to solve this problem and get the answer $$x=1$$.

Manipulation of Radicals
Let's say that we have a number $$x$$. The square root of $$x$$ is the number that, if multiplied by itself, equals $$x$$. 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 $$2 \times 2 = 4$$) or it could be -2 (because $$-2 \times -2 = 4$$). We use the symbol $$\sqrt{x}$$ to indicate the positive square root of $$x$$.

The cube root of $$x$$ is the number that, if multiplied by itself three times, equals $$x$$. We use the symbol $$\sqrt[3]{x}$$ to indicate the cube root of $$x$$.

We use the symbol $$\sqrt[n]{x}$$ to indicate the number, which when multiplied $$n$$ times is equal to $$x$$. Or in symbols: if $$y = \sqrt[n]{x}$$ then $$y^n = x$$.

Rules

 * 1) $${\sqrt{x}}\cdot{\sqrt{x}}=(\sqrt{x})^2={x}$$
 * 2) $$={{\sqrt{x}}\over{\sqrt{y}}}$$
 * 3) $$x^{{m}\over{n}}={({\sqrt[n]{x}})^m}$$
 * 4) $${\sqrt{x}}{\sqrt{y}}=$$
 * 5) $$==x^{1\over{m \cdot n}}$$

Solving for (Variable)
When solving an equation, you usually solve for a specific variable. To do so, you have to get all instances of that variable on one side of the equals sign, and everything else on the other.

Properties of Equality
The equal sign that depicts the fact that both sides of it are equal is a very strange symbol with many properties. It tells you various traits of each side, and it allows you to manipulate each side in specific ways. Here are the different properties of that sign:

Practice Problems
 {Decide whether these following problems are expressions or equations.}

{$$2x + 6$$ +expression -equation
 * type=""}

{$$3(x - 14)^2 = 12$$ -expression +equation
 * type=""}

{$$4 + 6(2x + 16)$$ +expression -equation
 * type=""}

{Identify which properties are being used in the following problems.}

{$$a = b$$ and $$3 - a = 4$$, so $$3 - b = 4$$. { Substitution (i) _12 } Property of { Equality (i) _8 }
 * type="{}" coef="2"}

{$$x + 9 = 12$$, then $$x = 3$$. { Subtraction (i) _12 } Property of { Equality (i) _8 }
 * type="{}" coef="2"}

{$$x - 9y = 4y$$, then $$x = 13y$$. { Addition (i) _12 } Property of { Equality (i) _8 }
 * type="{}" coef="2"}

Basic Laws In Algebra
In algebra we are working with the set of real numbers. We talked about the properties of real numbers with respect to mathematical operations in the section Real Numbers. If you don't remember the Commutative Property, the Associative Property, the Distributive Property, and the Identity Property go back to the real numbers section to review them.

There are several basic laws in algebra. Understanding these will help you to manipulate and solve equations, and to understand algebraic relationships.

Proportions or Ratios
Ratios or proportions can be expressed as an equation of fractions

(for example $$ \frac {Q}{R} = \frac {S}{T} $$ ),

or they can be expressed as a relationship Q : R = S : T, (expressed in words “ ‘Q’ is to ‘R’ as ‘S’ is to ‘T’ ”).

Using the words "is to" help us understand the physical relationship between the values, while the fractional representation can help us deal with the mathematical relationship.

For instance in America we know that 3 feet is to 1 yard as 6 feet is to 2 yards, but expressing this as a math equation: $$ \frac {3}{1} = \frac {6}{2} = \frac {2*3}{2*1} $$ helps us see this is true because all we have done is double the original ratio.

Consider the relation 3*4 = 2*6 = 1*12. Does this mean something more than the obvious fact that 12=12=12? How many ways could you package 12 items?

In general, these relationships can be described by a general equation like $$Q \cdot T = S \cdot R$$

Dividing each side of the equation by $$T \cdot R $$ gives

$$ \frac{Q \cdot T}{T \cdot R} = \frac{S \cdot R}{T \cdot R} $$, which simplifies to $$ \frac{Q}{R} = {\frac{S}{T}} $$

If we divide by $$Q \cdot S$$ instead the results simplify to

$$ \frac{T}{S} = {\frac{R}{Q}} $$

One needs to be careful since each term also has a proportional relation with the two other variables.

In our examples all of the following are also valid Q : S = R : T ,“ ‘Q’ is to ‘S’ as ‘R’ is to ‘T’ ”. R : Q = T : S ,“ ‘R’ is to ‘Q’ as ‘T’ is to ‘S’ ”.. S : Q = T : R ,“ ‘S’ is to ‘Q’ as ‘T’ is to ‘R’ ”.

Since 2*6=3*4 2:3 = 4:6, 2 is to 3 as 4 is to 6, and 2/3 = 4/6. 4:2 = 6:3, 4 is to 2 as 6 is to 3, and 4/2 = 6/3. 3:2 = 6:4, 3 is to 2 as 6 is to 4, and 3/2 = 6/4. 2:4 = 3:6, 2 is to 4 as 3 is to 6, and 2/4 = 3/6.

Solving Equations
Although we have already solved a few equations, we will now discuss the formal idea of solving equations. To solve an equation, you are finding the value of any variables within the equation. To find the value of a variable, you have to manipulate the equation and reach the state $$\text{variable} = \text{number}$$. Then you know the value of the variable! You will use the Properties of Equality to manipulate the equation into the desired form.

Practice Problems
 {Solve for x in the following equations.}

{$$x + 7 = 12 \,$$ $$x=$${ 5_17 }
 * type="{}"}

{$$-\frac{2}{3}x + 9 = 15$$ $$x=$${ -9_17 }
 * type="{}"}

{$$3x - 15 = 8x \,$$ $$x=$${ -3_17 }
 * type="{}"}

{$$c(x - 4m) = d(n - 9) \,$$ $$x=$${ (4cm + dn - 9d)/c (i)|(4cm+dn-9d)/c (i) _17 }
 * type="{}"}

{$$20x + 12 = 4(5x + 3) \,$$ $$x\in$${ R (i) _17 }
 * type="{}"}

Lesson Review
Equations are two expressions that are equal to each other, and they are expressed by putting each of them on one side of the equal sign. You can add, subtract, multiply, or divide both sides of an equation while keeping it equal (For example, we know that 7 = 7, correct? What if we subtracted 2 from each side? We'd still have a true statement: 5 = 5). There are other properties of equality, such as the Reflexive, Symmetric, Transitive, and Substitution. You will be using all of these properties to solve (find the value of) variables in equations.

Lesson Quiz
 {What property is expressed here? $$a = b$$ and $$b = c$$, then $$a = c$$. { Transitive (i) _10 } Property of { Equality (i) _8 }
 * type="{}" coef="2"}

{If I divided both sides of an equation by 4, would it still be equal on both sides? +yes -no
 * type=""}

{Why? { Division (i) _10 } Property of { Equality (i) _8 }
 * type="{}" coef="2"}

{Solve for y. $$5\left(\frac{y}{6} - 2 \right) = 2y + 4$$ y={ -12_3 }
 * type="{}"}

title: Algebra/Chapter 3/Solving Equations


text:
Algebra/Chapter 3/Word Problems

Word Problems are mathematical problems that are delivered in ordinary words, instead of mathematical symbols. Part of the problem with dealing with word problems that they first need to be translated into mathematical equations, and then the equations need to be solved.

Working with Word Equations
Math is a kind of language, with its own "grammar"; to become "fluent" in math, you must understand its grammar. Some of the "grammatic" terms used in Math include:

This is not a definitive listing. While problems found in text books usually include these phrasings, you may encounter one where such words are implicit.

Steps To Solving Word Problems
The majority of word problems in algebra can be rewritten as equations, which you can then solve. There are some steps to take as you analyze a word problem.

 * 1) Read Carefully
 * 2) Define A Variable (What You Are Looking For)
 * 3) Make A Plan
 * 4) Write An Equation
 * 5) Solve
 * 6) Is This Answer Reasonable?

These six steps may seem either obvious, but serve as the foundation to solving word problems.

Exercises
 {Steven bought 4 packs of baseball cards, each with the same number of cards inside. If Steven had 32 cards overall, how many cards were in each pack? { 8_10 }
 * type="{}"}

{4 barrels of equal volume are filled with water. 3 of the barrels are completely filled and the fourth barrel is filled with only 8 gallons. If 53 gallons of water have been pumped into the barrels overall, what is the volume of each barrel? { 15 gallons (i)|15gallons (i)|15 gallon (i)|15gallon (i)|15 gal (i)|15gal (i)|15_10 }
 * type="{}"}

{Jason pays each of his employees equally. If Jason had to pay each employee $50 and he gives them each a 10% bonus, how many employees does Jason have if he pays $440 overall? { 8_10 }
 * type="{}"}

{A car travels 20 km/h faster than a second car. The first car covers 180 km, the second car covers 135 km in the same time. What is the average speed of each car? s1={ 80 km/h (i)|80km/h (i)|80 kmh (i)|80kmh (i)|80 kph (i)|80kph (i)|80 kmph (i)|80kmph (i)|80_10 } s2={ 60 km/h (i)|60km/h (i)|60 kmh (i)|60kmh (i)|60 kph (i)|60kph (i)|60 kmph (i)|60kmph (i)|60_10 }
 * type="{}" coef="2"}

{Quinton and Thomas work at a company. Quinton's wages are $20 per hour, while Thomas has wages of $40 per hour. Quinton, however, being in sales, can expect a $3 raise (per hour) every month, while Thomas can only expect a raise of 50 cents. How long until they are paid the same amount per hour? { 8_10 } months
 * type="{}"}

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Algebra/Chapter 3/Word Problems


text:
Algebra/Chapter 3/Percent

3.7: Percentages

Percentages
Percentages are another way of representing rational numbers. Rational numbers can be represented as decimals, fractions or percentages.

Percent, when you actually break the word apart, consists of two words: per and cent. Per is a small but extremely powerful word. It means "to divide". Cent is a Latin word for 100. Percent, then, means "divide by 100".

To convert from a fraction to a percentage, we simply multiply the fraction by 100. For example, $$\frac{1}{4} = (\frac{1}{4} \times 100)\% = 25\%\,$$.

To convert from a decimal to a percentage, we first convert the decimal into a fraction, and then proceed with the approach outlined above. For example, $$0.15 = \frac{15}{100} = (\frac{15}{100} \times 100)\% = 15\%\,$$.

title: Algebra/Chapter 3/Percent


text:
Algebra/Chapter 4/Interval Notation

Interval notation and the Number Line
The real numbers can be represented on a number line, a line theoretically extending infinitely in two opposite directions as shown here:

The arrowheads at the opposite ends of the drawing of the number line mean that line in concept extends infinitely in those directions, even though the drawing of the line cannot be extended forever in those directions. Note that the right side of the number line stretches to positive infinity and the left side stretches to negative infinity. Numbers in a set can be shown as dots on (or near) a number line. For example, the above set of natural numbers from 1 to 8 would be shown as follows:

Often, a series of numbers will go on infinitely in one or both directions. For example, the set of natural numbers, consisting of numbers one naturally counts with, starts with 1, 2, 3, 4, and goes on to infinity. The indefinite continuation of an infinite set of numbers (or similar elements) can be written as several dots (called an ellipsis) after some numbers or elements listed showing the initial trend. Thus, the set of natural numbers can be represented as follows:

$$\lbrace1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ...\rbrace \,$$

where the three dots represent the continuing trend of an infinite set of elements The set of integers can be represented as follows:

$$\lbrace ..., -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, ...\rbrace \,$$

There is no particular requirement that the listing of an infinite set of elements stop at 10 or 8 or any particular number, as long as a clear, understandable trend is given. Intuitively we can the set of natural numbers is a subset of the set of integers when we find the number 1 in the integers and see that the set continues to count upward forever. So far, we have discussed discrete numbers. Discrete means consisting of one or more isolated, individual numbers or points; or not having a continuous range (interval) of numbers or points.

Between every two integers, there are an infinite number of fractional, or rational numbers. Furthermore, between any two fractional numbers, there are an infinite number of other fractional numbers, and so on. This characteristic is sometimes referred to as continuity. Such a continuous set of numbers is represented as a bold line segment on (or near) the number line, similar to the way a continuous set of points is represented by a line segment in geometry. A continuous set of numbers which includes all the numbers between two given numbers is often called an interval. The two numbers that the continuous set of numbers are between are the endpoints of the line segment. One, both, or neither of the numbers at the endpoints of the interval may be included with the set of numbers in the interval. If the number at the endpoint is included, that endpoint is a closed endpoint and is represented by a solid dot. If the number at the endpoint is not included, that endpoint is a open endpoint and is represented by a hollow dot (a tiny hollow circle). As an example, shown below on a number line is the interval between 1 and 8 which includes 1 (is closed at 1) but does not include 8 (is open at 8):

To save ourselves time describing these intervals we often represent them with two types of parentheses, [ ] and. The square brackets, [ ], mean an inclusive interval- that is, the numbers inside the brackets are included, much like a solid point on our number line interval. The rounded brackets,, mean an exclusive interval- that is, the numbers inside the brackets are excluded from the interval, much like the hollow point on the number line. We can also use the symbol $$ \ge $$ to show that a statement is inclusive on the left and the symbol $$ \le $$ to show it is inclusive on the right. The symbols $$ > $$ and $$ < $$ show that the interval is exclusive. Consider these examples where we let X represent any number on the interval:

$$[4, 9] = 4 \le X \le 9 \,$$ or all numbers from 4 to 9.

$$(4, 9) = 4 < X < 9 \,$$ or all numbers between 4 and 9.

$$[4, 9) = 4 \le X < 9 \,$$ or all numbers between 4 and 9 including 4.

As with the third example, we can use combinations of the two types of parentheses to display any interval on the real numbers. A set of continuous numbers can also be defined which starts (or ends) at one number and extends infinitely in either the positive direction or the negative direction. Geometrically, such a set is represented by a ray on the number line, where the continuous set of numbers is shown as a bolder part of the line. If the endpoint is included in the set, the endpoint is closed and represented by a solid dot. If the endpoint is not included in the set, the endpoint is open and represented by a hollow dot. As an example, a set of numbers greater than or equal to 1, is shown on a number line below:

This is equivalent to the interval $$[1, \infty]$$. In another example, a set of numbers less than 8, $$[-\infty, 8)$$, is shown on a number line below:

A set which contains all the solutions to an algebraic equation is called that equation's solution set, i.e. all the numbers that if substituted for an "unknown" variable in that equation would make it true. A formula is a math "process" that finds an answer to different unknown variables by using other variables and numbers. An example of a formula is Einstein's formula: $$E = Mc^2 $$; if you know the mass of an object, M, and you multiply it by the speed of light squared ($$c^2$$), you get its energy, E. Formulae like these can be rearranged to find the values of different variables, too.

Conceptual Questions
 Problem 4.1 (Explaining Inequalities)

 Problem 4.2  Suppose two values A and B are non-equal, can you determine which of the two quantities below is larger?

Quiz
 {Name whether each expression would be represented by a number line, line segment, or ray. If the answer is a line segment or ray, identify each end of the segment and whether the non-infinite ends would be represented by a filled or hollow dot.}

{ > 8 -line -line segment +ray
 * type=""}

{The ? bound is equal to 8. +lower -upper
 * type=""}

{The dot would be +filled -hollow
 * type=""}

{< 5 > +line -line segment -ray
 * type=""}

{< -4 -line -line segment +ray
 * type=""}

{The ? bound is equal to -4. -lower +upper
 * type=""}

{The dot would be -filled +hollow
 * type=""}

{ > 2 < 7 -line +line segment -ray
 * type=""}

{The ? bound is equal to 2. +lower -upper
 * type=""}

{The dot would be +filled -hollow
 * type=""}

{The ? bound is equal to 7. -lower +upper
 * type=""}

{The dot would be -filled +hollow
 * type=""}

{Identify a probable next value in the set, given the values shown.}

{ {{ 0_5 }, 1, 2, 3, { 4_5 }}
 * type="{}" coef="2"}

{ {3/4, 4/4, 5/4, { 6/4|3/2|1 2/4|1 1/2|1.5_5 }}
 * type="{}"}

{ {{ -4_5 }, -2, 0, 2}
 * type="{}"}

{How would you represent the following number sets, assuming that sides of the number set which aren't explicitly closed continue forever? (use  for   and   for  )}

{{..., 9, 10, 11} { <= 11 (i)|=< 11 (i)|<=11 (i)|=<11 (i) _8 }
 * type="{}"}

{{9, 10, 11, ...} { >= 9 (i)|=> 9 (i)|>=9 (i)|=>9 (i) _8 }
 * type="{}"}

{{8, 9, 10, 11, 12, 13} { >= 8 (i)|=> 8 (i)|>=8 (i)|=>8 (i) _8 }, { <= 13 (i)|=< 13 (i)|<=13 (i)|=<13 (i) _8 }
 * type="{}" coef="2"}

title: Algebra/Chapter 4/Interval Notation


text:
Algebra/Chapter 4/Inequalities

Definition of 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 real life we often use inequalities more than equations (i.e. this shirt costs $2 more than that one).

Given that a and b are real numbers, there are four basic inequalities:

 * 1) a < b --> a is "less than" b
 * Example: 2 < 4 ; -3 < 0; etc.
 * 1) a > b --> a is "greater than" b
 * Example: -2 > -4 ; 3 > 0 ; etc.
 * 1) $$a \le b$$ --> a is "less than or equal to" b
 * Example: If we know that $$x \le 7$$, then we can conclude that x is equal to any value less than 7, including 7 itself.
 * 1) $$a \ge b$$ --> a is "greater than or equal to" b
 * Example: Conversely, if $$x \ge 7$$, then x is equal to any value greater than 7, including 7 itself.

Possible Relationships
A number on the number line is always greater than any number on its left and less than any number on its right. The symbol "$$<$$" is used to represent "is less than", and "$$>$$" to represent "is greater than".

Consider this number line: From the number line, we can easily tell that 3 is greater than -2, because 3 is on the right side of -2 (or -2 is on the left of 3). We write it as $$3 > -2$$. A teacher in elementary school told me to think of the greater sign as a "greedy mouth". The mouth always tries to eat the bigger number. When I look at the number line I can see that the greater and lesser signs could also represent the arrows at the ends of the number line. In this case the inequality signs represent arrows that point which direction I have to go to get to the first number from the second number. If the second number is smaller than the first than I have to move to the right on the number line to go from the second number to the first (which is why the "mouth" points toward the first number). Similarly if the second number is larger than the first then I have to move to the left on the number line (and the "mouth" now points toward the second number). I could never remember my teacher's rule of thumb for the "greedy mouth", but when I thought of the sign as the arrow heads at the end of the line the direction of the operator made sense to me. How are you going to remember that "$$<$$" is used to represent "is less than", and "$$>$$" to represent "is greater than"?

Consider a number $$a$$ and a constant $$C$$. One and only one of the following statements can be true:
 * 1) $$a > C$$
 * 2) $$a = C$$, or
 * 3) $$a < C$$

This is the Law of Trichotomy. Another way of describing it is that given a fixed number and a variable the variable can represent numbers smaller than the number, at the same point as the number, or that are larger than the number. When we create inequalities from words using symbols we know that only one of these statements can be true.

 Word Problems

Properties
There are four important properties with inequalities: 1. Transitive property: For any three numbers $$x$$, $$y$$ and $$z$$, if $$x > y$$ and $$y > z$$, then $$x > z$$.

2. In an inequality, we can add or subtract the same value from both sides, without changing the sign (i.e. "$$>$$" or "$$<$$"). That is to say, for any three numbers $$x$$, $$y$$ and $$p$$:
 * if $$x > y$$, then $$x + p > y + p$$ and $$x - p > y - p$$.

3. We can multiply or divide both sides by a positive number without changing the sign. For example, if we have any two numbers $$x$$ and $$y$$, and another positive number $$p$$:
 * if $$x > y$$, then $$x \times p > y \times p$$ and $$\frac{x}{p} > \frac{y}{p}$$.

4. When we multiply or divide both sides by a negative number, we have to change the sign of the inequality (i.e., "$$>$$" changes to "$$<$$" and vice versa). So if we are given two numbers $$x$$ and $$y$$, and another negative number $$p$$
 * if $$x > y$$, $$x \times p < y \times p$$ and $$\frac{x}{p} < \frac{y}{p}$$.

Now we can go on to solve any linear inequalities.

Solving Inequalities
Solving inequalities is almost the same as solving linear equations. Let's consider an example: $$x + 4 < 13$$. All we have to do is subtract 4 on both sides. We will then get $$x < 9$$, and that is the answer! Note, however, that what you get is not a single answer, but a set of solutions. Any number that satisfies the condition $$x < 9$$ (any number that is less than 9) is a solution to the inequality. It is very convenient to represent the solution using the number line: <---o <-+-+-+-+-+-+-->  6     7     8     9     10    11

Note: the circle(o) shows that the value 9 is not included. Later on, when we deal with less than or equal to and greater than or equal to (≤ and ≥), we use "*" to show that the value is included in the solution set.

Let's try another more complicated question: $$3x-2 > 2(x-3)$$. First, you may want to expand the right hand side: $$3x-2 > 2x-6$$. Then we can simply rearrange so that all the unknowns are on one side (usually the left): $$3x-2x > -6+2$$. Hence, we can easily get the answer: $$x > -4$$.

Here's an example where the direction of the inequality changes when finding the solution: solve $$4 - 6x > 22$$. So the solution to the inequality is $$x < -3$$.
 * First subtract 4 from both sides: $$-6x > 18$$.
 * Now divide through by -6, changing the direction of the inequality: $$\frac{-6x}{-6} < \frac{18}{-6}$$.

Inequalities, unlike equations, commonly have an infinite number of solutions.

$$x > A \!\ $$ means that x is greater than A

$$x < A \!\ $$ means that x is less than A

Special Cases - Multiplying by -1
The rule for multiplying or dividing each side of an inequality by a negative number states that you also need to change the direction of the inequality. Look at the inequality -1 < 1. Let's rewrite this generically as -1 op 1, although we know that the operator should be <. Now multiply both sides of the inequality by -1. You get (-1)*(-1) op (1)*(-1). Simplifying you get 1 op -1. For this statement to be true the operator now needs to be >.

If we wanted to we could change multiplying by a negative number into two operations: [(-1)*(Number)][-1 < 1][(Number)*(-1)]. This may seem obvious since we know that all we have to do to get a number to turn into its opposite is multiply by -1. But, what is it about multiplying by -1 that causes us to have to change the direction of the inequality operator? If you represent multiplication by the number 1 on a number line you see that all you do is move x(the number you are multiplying) units to the right of zero.

 TODO: Need Graphic

Similarly, if you represent multiplication by the number -1 on a number line you see that you need to move x(the number you are multiplying) units to the left of zero.

 TODO: Need Graphic

Now think about the difference between an equality and an inequality. If you know that x = y is true then you also know that x-y = 0. If you know that x > y then you know two things: that x - y > 0 (bigger - smaller is positive) and also that y - x < 0 (smaller - bigger is negative). If we multiply x > y by 1 we still get x > y. If we move y to the left side of the equation by subtracting y from both sides we get x - y > 0. Which we know is true. On the other hand when we multiply x > y by -1 and don't change the direction of the inequality we get -x > -y. When we move the y to the left side of the equation by adding y to both sides we get y - x > 0. Which we know is not true. In the previous step when we needed to say -x < -y because even though both numbers are negative, -x is further to the left, and therefore smaller, than -y.

Working algebra problems can become automatic, but it is important for you to keep in mind why the rules are true so that you can catch yourself if you accidentally forget part of a rule.

Special Cases - Inequalities with a variable in the denominator
For example, consider the inequality
 * $$\frac{2}{x-1}<2\,$$

In this case one cannot multiply the right hand side by (x-1) because the value of x is unknown. Since x may be either positive or negative, you can't know whether to leave the inequality sign as <, or reverse it to >. The method for solving this kind of inequality involves four steps:
 * 1) Find out when the denominator is equal to 0.  In this case it's when $$x=1$$.
 * 2) Pretend the inequality sign is an = sign and solve it as such: $$\frac{2}{x-1}=2\,$$, so $$x=2$$.
 * 3) Plot the points $$x=1$$ and $$x=2$$ on a number line with an unfilled circle because the original equation included a < sign (note that it would have been a filled circle if the original equation included <= or >=).  You now have three regions which are separated by unfilled circles. These regions are: $$x<1$$, $$12$$.
 * 4) Test each region independently.  in this case test if the inequality is true for 12 (e.g. x=3).  In this case the original inequation holds, and so the solution for the original inequation is x>2.

Special Cases - Going back to equality
So far, you have seen various kinds of inequalities. However, there are still two kinds of inequalities that we must discuss:

 * 1) $$x \ge A$$ means that x is greater than or equal to A;
 * 2) $$x \le A$$ means that x is less than or equal to A.

These kinds of inequalities also include the possibility of $$x = A$$. In most cases, they can be obtained if we negate simple inequalities like $$x > A$$ or $${x < A}$$. For example, consider the inequality

$$-5x > 3$$

To solve this inequality, we have to divide both sides by -5 (thus negating the inequality and isolating x). Notice that the right hand side (RHS) of the inequality will be negative ($$-\frac{3}{5}$$) now. So we must flip the sign.

In essence, to flip the $$>$$ sign, just replace it in the inequality with $$\leq$$ or vice versa. On the other hand, in order to flip the $$<$$ sign, replace it with $$\geq$$ or vice versa. Keeping all this in mind, we can continue solving the inequality above:

$$x > {-\frac{3}{5}}$$. After dividing both sides by -5, notice that we have an isolated x and a negative RHS. Flipping the sign gives:

$$x \leq -\frac{3}{5}$$. Which is the final answer!

Anyway, if you encounter inequalities with negative numbers, try to divide the whole inequality by the number next to x (so as to isolate x), flip the sign, and simplify.

Of course there are harder examples, where there are inequalities with multiple special cases. So, here's a tip: if you have a negative ratio with x in the denominator, just write a little line below the sign (since it gets inverted twice, due to negation and inversion). Just make sure to apply the negative sign to both sides first and then invert both sides after.

title: Algebra/Chapter 4/Inequalities


text:
Algebra/Chapter 4/Compound and Absolute Value Inequalities

Compound Inequalities are two inequalities presented as one problem. The variable being solved for may need to only fulfill only one of the inequalities, or both, depending on if the word "and" or "or" is used. There may be many correct answers, or even none at all.

Example: 2x - 4 > 2 or 30/x > 15

 * 2x > 6 or 30 > 15x
 * x > 3 or x < 2

The answer can then be graphed as two rays on a number line.

If we are interested in real solutions we can pick numbers barely smaller 2 or hardly greater than 3 as we want and the equation will be true. For instance if x =1.99 then 30/x = 15.08 (satisfies 30/x > 15) or if x = 3.01 then 2x-4 = 2.02 (satisfies 2x -4 > 2).

If we are only interested in integer solutions than we can see that all the integers except for 2 or 3 will work.

We can also express a compound inequality as a function.

If we specify a function f(x) = (x > 3 or x < 2) we get a graph like this:

The solution is still two rays. But now the rays are on the line y=x, with one ray moving to the right and up from the number 3 and the other moving left and down from the number 2.

Exercise: Using the concept of slope, can you explain why a figure transferred from a number line to a Cartesian graph will always be displayed on the line y=x?

Example: Highway and City Gas Mileage
The gas mileage sticker on cars gives two numbers: one for city driving, and one for highway driving. If a sticker says the car gets 25 mpg in the city and 32 mpg on the highway how far can you drive?

In order to find this out I need to know how much gas I have in my tank. I know that I can drive between 25 and 32 miles with a single gallon of gas. I can represent this with two equations where x = how far I can drive on a gallon of gas:

25 < x and x < 32.

Or I can combine them into a compound inequality:

25 < x < 32.

If I have a 10 gallon gas tank, and I want to know how far I can drive I change the equation to represent the 10 gallons of gas I have in my tank by multiplying each term in the compound inequality by 10.

25 * 10 < 10*x < 32 * 10.

250 < 10x < 320.

I can drive between 250 and 320 miles on a single tank of gas.

Example: Conversions
The conversion formula for Fahrenheit to Celsius is 5/9(F - 32).

We know that water freezes at 32 degrees Fahrenheit and boils at 212 degrees Fahrenheit.

If we want to know the freezing point and boiling point of water in Celsius we could set the equation up as follows:

32 < F < 212

And then apply the formula to convert the values in degrees F to degrees C.

5/9(32-32) < C < 5/9(212 - 32)

So water freezes and boils at:

0 < C < 100

Absolute Value
What happens when we need to multiply by x?

Remember that subtraction is equivalent to adding a negative number, and multiplication is the same as adding a number multiple times. So when we add a negative number multiple times the result keeps getting smaller. When solving a problem using inequalities we need to decide if the context of the problem makes sense for negative numbers.

A special type of compound inequality involves the absolute value sign. The absolute value sign specifies two functions such that f(x)=A implies f(-x) = (-A)(-1). For the equal sign this is the same as f(x)=f(-x)=A. But when we use an inequality sign and we multiply by a negative number we have to switch the direction of the sign. So if f(x) > A then f(-x) < A.

This is simple when the only the value of x is in the absolute value sign. For instance given|x| > 12 then the possible solutions are x > 12 or x < -12.

Example - convert |6x + 18| - 33 > 9 into a compound inequality.

 * 6x + 18| > 36
 * 6x + 18 > 36 or 6x + 18 < -36
 * 6x > 18 or 6x < -54
 * x > 3 or x < -9

How can you use absolute value?
Example 1

A good way to use absolute value is to determine if your previous estimates are right.

For instance, given your car that gets between 25 and 32 miles per gallon. If you want to make sure my car is running well you should keep track of my mileage using the formula:

$$25 < \left | m / g \right \vert < 32 \ $$ when m = (miles at last fillup - miles at the pump) and g = amount of fuel I just pumped.

If m/g is not greater than 25 than you should make sure your car is running properly. If m/g is over 32 maybe someone is putting extra gas in your car!

Example 2 - Think like a teacher!

You give five tests in a quarter. A student has made scores of 80, 90, and a 80 on the first three tests. The student asks you for a chance for extra credit. Should you give the student the chance?

To get a B for the quarter the student needs to make 80*5 = 400. To get an A for the quarter the student needs to get a 450. If we let X equal the student's scores on the last two tests we can see

400 < 250 + x < 450

150 < x < 180

The average score for the last two tests will be x/2. So dividing the equation above by 1/2 we get 75 < x < 90. Since the average score until now is about 83 the student would be encouraged to study harder for the final two tests with just a few extra credit points.


text:
ID Board Review/Bacterial Diseases/Mycobacterium/Mycobacterium tuberculosis/TB Mycobacterium

 * Also commonly referred to as "tuberculosis," or "TB" for short.


text:
ID Board Review/Bacterial Diseases/Mycobacterium/Mycobacterium tuberculosis complex/Mycobacterium leprae

Clinical Presentations

 * Leprosy


text: User talk:Nahian

title: User talk:Nahian


text:
User talk:2601:240:E000:FB10:A5C1:7740:7BA5:A4B

Please try to improve the page The Lingo Show/Words or it will be deleted in the next day or two. A list of words in other languages is not useful. This is Wikibooks where books are created. Please stop adding lists of words and make the book more useful. Stop adding more lists of words to the page and fix it first.

Recent edits
A page you have edited, The Lingo Show/Words, has been deleted. It was queried as not suitable for Wikibooks on the day it was created, 10 days ago. It is not suitable for Wikibooks because it appears to be nothing more than a list of words with no educational value. It was also not formatted as a book. Numerous attempts were made to suggest to the editors of the page that improvements should be made but editors continued to add more words to the long lists. The page has now been deleted. If you wish to contribute more please look at existing books on the project to see what typical Wikibooks books look like and also look at Using Wikibooks. --Xania   talk 11:13, 24 April 2022 (UTC)


text:
Minecraft/Villager

{{minecraft/mob
 * spawning=In villages. Doesn't respawn.|cat=Humanoid|health=20

 * intro=Villagers are found in villages and trade for and with emeralds.

 * des=These mobs are found in villages and can have several professions. Each villager must have a bed to sleep the nights in, and destroying a workstation in a village causes the villager employed at it to become unemployed and start looking for a new job.

Villagers can open doors.

If you hold an item, nearby villagers who want the item will approach and show you what they have to offer. This is called "trade offering".

Each profession has their own items to buy and sell, and each individual within the profession sell different amounts, and each have different prices.

Each time you complete a trade, both you and the villager gain Exp. points. If the villager levels up, it will unlock new trades. Each villager may only level up 5 times. You can check a villager's level by looking at the badge on their leg.

 * Lv 1, novice, stone (gray) badge
 * Lv 2, apprentice, iron (white) badge
 * Lv 3, journeyman, gold (err... gold) badge
 * Lv 4, expert, emerald (green) badge
 * Lv 5, master, diamond (blue) badge

After you trade a single trade enough times, it becomes unusable. After a few minutes, if they have a bed and can get to their workstation, it will become available again, but the price will rise. If you wait a day without trading it so much that it gets locked, the price should go back down, assuming it can still reach its bed and workstation.

There are two ways for a trade to go down:
 * 1) Have the hero of the village buff. This can be obtained by defeating a raid.
 * 2) If the villager was previously a zombie villager and was cured, or saw a villager get cured, it will be thankful and offer a permament discount for all trades.

It is possible to get unlimited emeralds. If you can make a villager librarian offer a bookshelf for one emerald, buy it, place and break the bookshelf, then sell the resulting books back to the same librarian, you are basically getting free emeralds. All you need to start with is one emerald.

Breeding
Villagers are their own people. You can't just force them to have a baby. In order for two villagers to have a baby, 3 requirements must be met:
 * 1) There is at least one more bed than villagers within the village
 * 2) Both villagers must have enough food (just throw 16 peices of bread or 25 carrots / potatoes on the ground in front of two different villagers)
 * 3) You remain in or near enough to the village that you can still see the villagers. If you get too far, they will stay in one place and not move or breed. See how the game works: spawning.

For more on ../Zombie Seige/s and ../Pillager Raid/s, see their respective pages.

 * end=

Achivements
}}

title: Minecraft/Villager


text:
User talk:2600:1008:B040:9261:C13B:E530:4808:8F4C

You cannot edit for 6 hours due to disruptive editing. You are welcome to contribute constructively when editing privileges are restored. You may appeal this action with a reasonable argument by adding the text below.

--Xania   talk 13:46, 22 April 2022 (UTC)


text:
Minecraft/Zombie Seige

A zombie siege is when zombies spawn in large quantities within a village. The only variant of zombie that can spawn is a baby zombie.

Zombies that spawn during the event behave more aggressively than other zombies and will ignore sunlight.

Every night, at midnight, if you are in a village, there is a 1 in 10 chance of one starting. Once it starts, 20 zombies will spawn within the village and start to attack the villagers.

Lighting up the village perfectly and on every block, and surrounding it with a wall will not protect the village from a siege.

title: Minecraft/Zombie Seige


text:
User talk:204.116.100.123

--Xania   talk 14:15, 22 April 2022 (UTC)

 * yes i will 204.116.100.123 (discuss) 15:13, 22 April 2022 (UTC)

title: User talk:204.116.100.123


text:
Talk:Transportation Systems Casebook

Thoughts on the existing page and discussion on addition of adding content for: 2020 cases 2021 cases 2022 cases

Addition to Case Timeline
Adding content for: 2020 cases 2021 cases 2022 cases Shpookie (discuss • contribs) 14:57, 22 April 2022 (UTC)

title: Talk:Transportation Systems Casebook