Ch+1+0910

​Editor: Haley Brandt Notes Sections 4-7:Katie Conner Example Problems: Matthew Aguilar Definitions: Austin Gibbons Note sections 1-3: Cheyenne Bohlen

__1.1 Real Numbers and Number Operations:__ Real numbers can be pictured as points on a line called a real number line. <-0> =the zero is the origin.= Properties: closure- a + b= real number. ab= real number commutative- a+b= b+a ab=ba associative- (a+b) +c= a+(b+c) (ab)c= a(bc) identity- a+0=a,0+a=a inverse- a+(-a)=0 distributive- a(b+c)=ab +ac

__1.2 Algebraic Expressions and Models__ numerical expressions=consist of numbers, operations, and grouping symbols. evaluating expressions- 1. do operations inside grouping symbols. 2. evaluate powers. 3. multiply and divide. 4. add and subtract.

__ 1.3 __ an equation is a statement where two expressions are equal. a linear equation can be written as ax=b.

Important Formulas to know!

 * = To Find: ||= Use the Formula: ||= Variable Meanings: ||
 * = Distance ||= d=rt ||= D=distance R=rate T=time ||
 * = Simple Interest ||= I=Prt ||= R=rate I=interest P=principal T=time ||
 * = Temperature in Fahrenheit ||= F=9/5C+32 ||= F=Degrees Fahrenheit C=degrees Celsius ||
 * = Area of a Triangle ||= A=1/2bh ||= A=area B=base H=height ||
 * = Area of a Rectangle ||= A=lw ||= A=area L=length w=width ||
 * = Perimeter of a Rectangle ||= P=2L+2W ||= P=perimeter L=length W=width ||
 * = Area of a Trapezoid ||= A=1/2(b1+b2)h ||= A=area b1=base H=height b2=base 2 ||
 * = Area of a circle ||= A=(pi)rˆ2 ||= A=area Pi=3.14 R=radius ||
 * = Circumference of a circle ||= C=2(pi)r ||= C=circumference Pi=3.14 R=radius ||

3.Add: 10
= =

=__ 1.5 Problem Solving Using Algebraic Models: __=

One thing you have discovered is the approximate exchange rate between the American dollar and the foreign bill that is used in the country.
====If eight American dollar equals 87 times the foreign tender, you would then have to calculate how many American dollars it would require to purchase a plane ticket that costs 696 of the foreign currency.==== ====__Explanation__: To set up the equation, you must label each piece of information that is given. The exchange rate is 8:87. This represents the function, (//f//). The amount of American dollars that you would need to use is represented by the variable (//x//). The total cost in the foreign currency (696) is final represented by (//y//). Because you know all the variables' information except for (//x//), it is concluded that (//x//) is the variable to be solved for. The first step of setting the equation up is finding out which variables are related. Because the exchange rate between American to Foreign currency is 8 to 87, the two variables (//f//) and (//x//) must be decided to be related. That is the first half of the equation. The other half is the total of the desired plane ticket, (//y//). So the final equation would be (//f//)(//x//) = (//y//), or (87/8)x = 696. The reason (87/8) is used is because the exchange rate is being used in reverse, where you are not using the rate by multiplying it by the foreign currency to gain the total American amount, not initially at least. From here you can solve equation to discover how many dollars are needed to pay. The first step is to obviously multiply each side by (8/87) because it is the reciprocal of (87/8), which will allow you to get the variable (//x//) alone. The result of multiplying 696 by (8/87) is 64. So the final result of the equation is (//x//) = 64. This means it will require $64.00 to purchase a plane ticket out of the country. ​ ====

Definitions:
**Verbal Model** - a word equation that represents a real-life problem
 * Algebraic Expression** - a mathematical statement that represents a real-life problem

= __1.6 Solving Linear inequalities:__   =

1< x < 5 ß ---○**---|---|---|---○**--- à

…… 1 … 2 … 3 … 4 … 5 ====The numbers from the equation are marked with open circles because the sign doesn’t indicate that they are solutions to the problem. The numbers between the circles are bolded because they are all solutions to the equation.====

x __>__5 or x __<__ 2 ß ---|---•---|---|---•---|--- à …… 1 … 2 … 3 … 4 … 5 … 6  The numbers from the equation are marked with closed circles because the sign indicates that they are solutions to the problem. The numbers above 6 and below 2 are also bolded because they could be possible solutions to the problem.
 * “Or” Inequalities: **

====Assume that you are responsible for estimating how much distance a jet will be able to cover from a single fill up. ==== You know that after several testing experiences that in cold, moist conditions with high turbulence, the distance is brought down to 264 miles per fill. In addition to this knowledge, you also know that in dry, warm conditions with the assist of a tailwind, the mileage hits its peak at 747 miles. The best way to represent this knowledge is through a compound inequality. Only a single variable will be used due to the nature of the problem.

__Explanation__: First you must start off by assessing what information will go where. The minimum mileage is 264. This will be placed as the first third of the inequality. The maximum is 747 miles so this will be placed as the last third of the inequality. The final piece is represented by (x) because it is the mileage that is expected in neither of the other conditions. The way to set up the in equality is 264 < x < 747. "Less than or equal to" is used because the numbers used on each end are maximum and minimums that were replicated in tests, so they are possibilities in the mileage. This inequality can tell you how many fill-ups will be needed on a given destination if the flying conditions are also given. This is done by assessing the flying conditions to get the expected mileage, and then compare that mileage to the total destination distance to determine the number of fill-ups needed.

Definitions:
A//x// + B//y// › C, or A//x// + B//y// (greater than or equal to sign) C = __1.____7 Solving Absolute Value Equations and Inequalitie __ __s:__  =
 * ​Linear Inequality in one variable** - an inequality such as 2n - 3 › 9, and the inequality symbol is placed between two expressions
 * Linear Inequality in two variables** - an inequality that can be written in one of the following forms: A//x// + B//y// ‹ C, A//x +// B//y// (less than or equal to sign) C,
 * Solution of an Inequality in one variable** - a value of the variable that makes the inequality true
 * Solution of an Inequality in two variables** - an ordered pair (x,y) that, when x and y are substituted in the inequality, gives a true statement
 * Compound Inequality** - two simple inequalities joined by "and" or "or"
 * Graph of an Inequality in one variable** - all points on a real number line that correspond to solutions of the inequality
 * Graph of an Inequality in two variables** - the graph of all solutions of the inequality

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