Algebra 1 - Trimester 1 Study Guide
Algebra 1 - Trimester 1 Study Guide
Numbers & Operations
Starting with the WHOLE NUMBERS (0, 1, 2, 3, 4, ...) , also known as the counting numbers or natural numbers, we learned that adding ZERO and NEGATIVE NUMBERS to the whole numbers forms the set of INTEGERS. We defined the RATIONAL NUMBERS as the set of numbers formed from ratios of integers (fractions). We saw that rational numbers can be converted to their DECIMAL representations by dividing numerator by denominator. This decimal either terminates (ends) or contains a pattern of digits that repeats infinitely. We learned techniques for converting decimal representations (both terminating and repeating) to fractions and converting fractions to decimal representations and correctly rounding the result. We saw examples of numbers that could not be represented as a ratio of integers (2, π) and defined the set of IRATIONAL NUMBERS as the set of such numbers. The set formed by combining the rational numbers and the irrational numbers is called the set of REAL numbers. The course sequence traditionally taught in high school and early college (Algebra 1, Geometry (the analytic geometry part), Algebra 2, Precalculus and Calculus is essentially the study of sets of real numbers, functions of real numbers and how they can be used to model aspects of the world.
Once we developed an understanding of numbers we explored ways to combine them and make new numbers with the operations of ADDITION, SUBTRACTION, MULTIPLICATION and DIVISION. We did this by hand and using calculators. We saw that it is often useful to think about quantities as parts of one hundred. This informed our study of PERCENTS.
Starting with the WHOLE NUMBERS (0, 1, 2, 3, 4, ...) , also known as the counting numbers or natural numbers, we learned that adding ZERO and NEGATIVE NUMBERS to the whole numbers forms the set of INTEGERS. We defined the RATIONAL NUMBERS as the set of numbers formed from ratios of integers (fractions). We saw that rational numbers can be converted to their DECIMAL representations by dividing numerator by denominator. This decimal either terminates (ends) or contains a pattern of digits that repeats infinitely. We learned techniques for converting decimal representations (both terminating and repeating) to fractions and converting fractions to decimal representations and correctly rounding the result. We saw examples of numbers that could not be represented as a ratio of integers (, π) and defined the set of IRATIONAL NUMBERS as the set of such numbers. The set formed by combining the rational numbers and the irrational numbers is called the set of REAL numbers. The course sequence traditionally taught in high school and early college (Algebra 1, Geometry (the analytic geometry part), Algebra 2, Precalculus and Calculus is essentially the study of sets of real numbers, functions of real numbers and how they can be used to model aspects of the world.
Once we developed an understanding of numbers we explored ways to combine them and make new numbers with the operations of ADDITION, SUBTRACTION, MULTIPLICATION and DIVISION. We did this by hand and using calculators. We saw that it is often useful to think about quantities as parts of one hundred. This informed our study of PERCENTS.
2
Once we developed an understanding of numbers we explored ways to combine them and make new numbers with the operations of ADDITION, SUBTRACTION, MULTIPLICATION and DIVISION. We did this by hand and using calculators. We saw that it is often useful to think about quantities as parts of one hundred. This informed our study of PERCENTS.
Expressions
After introducing the notion of VARIABLE and the common practice of using letters to represent variable quantities we learned to create EXPRESSIONS using variables, numbers and operations. We practiced EVALUATING expressions by replacing a variable with a specific numerical value (and simplifying the result) and SIMPLIFYING expressions by combining like terms.
After introducing the notion of VARIABLE and the common practice of using letters to represent variable quantities we learned to create EXPRESSIONS using variables, numbers and operations. We practiced EVALUATING expressions by replacing a variable with a specific numerical value (and simplifying the result) and SIMPLIFYING expressions by combining like terms.
In Mathematica the built - in function Simplify is helpful.
In Mathematica the built - in function Simplify is helpful.
In[]:=
Simplifyn-2-2n+
3
2
3
2
Out[]=
-3+
11n
2
Expressions can be evaluated using the built-in function ReplaceAll
Expressions can be evaluated using the built-in function ReplaceAll
In[]:=
ReplaceAll[3w^2-3w+8,w->4]
Out[]=
44
In[]:=
ReplaceAll[10w+10x,{w->8,x->6}]
Out[]=
140
Expressions to Equations
Expressions joined by an equal sign (=) form EQUATIONS. To solve equations of a single variable we follow these steps: 1. Simplify both sides of the equation. (This might include using the DISTRIBUTIVE PROPERTY to remove parentheses) 2. Get the variable terms to one side of the equation and numbers to the other by adding to or subtracting from both sides of the equation. 3. When you have a single instance of a number multiplying a variable on one side of the equation and a single number on the other side multiply both sides of the equation by the RECIPROCAL of the number multiplying the variable (the COEFFICIENT). Equivalently you may divide both sides by this coefficient. 4. Check your solution by using it to evalute the original equation. In Mathematica the built-in function Solve can used for equation solving. Example: solve for x: 3(4x + 1) - 4x + 3 = -18 is solved with Mathematica as follows:
Expressions joined by an equal sign (=) form EQUATIONS. To solve equations of a single variable we follow these steps: 1. Simplify both sides of the equation. (This might include using the DISTRIBUTIVE PROPERTY to remove parentheses) 2. Get the variable terms to one side of the equation and numbers to the other by adding to or subtracting from both sides of the equation. 3. When you have a single instance of a number multiplying a variable on one side of the equation and a single number on the other side multiply both sides of the equation by the RECIPROCAL of the number multiplying the variable (the COEFFICIENT). Equivalently you may divide both sides by this coefficient. 4. Check your solution by using it to evalute the original equation. In Mathematica the built-in function Solve can used for equation solving. Example: solve for x: 3(4x + 1) - 4x + 3 = -18 is solved with Mathematica as follows:
In[]:=
Solve[3(x+1)-4x+3==18,x]
Out[]=
{{x-12}}
Absolute Value and Absolute Value equations
The absolute value of a real number is it’s distance from the origin and represented with vertical lines. |x| is read: “ the absolute value of x”. The absolute value equation |x| = a is equivalent to the question: what number(s) lie a units from zero? In this case there are two answers: a and -a. With numbers: |x + 4| = 7 is solved by remembering that there are two numbers 7 units from zero: 7 and -7 so the absolute value equation is broken into two equations: x + 4 = 7 or x + 4 = -7. Each equation is solved and the answer is given, x = 3 or x = -11. In Mathematica the Solve function can again be used (Abs[] is the built-in function for finding absolute value):
The absolute value of a real number is it’s distance from the origin and represented with vertical lines. |x| is read: “ the absolute value of x”. The absolute value equation |x| = a is equivalent to the question: what number(s) lie a units from zero? In this case there are two answers: a and -a. With numbers: |x + 4| = 7 is solved by remembering that there are two numbers 7 units from zero: 7 and -7 so the absolute value equation is broken into two equations: x + 4 = 7 or x + 4 = -7. Each equation is solved and the answer is given, x = 3 or x = -11. In Mathematica the Solve function can again be used (Abs[] is the built-in function for finding absolute value):
In[]:=
Solve[Abs[x+4]==7,x]
Out[]=
{{x-11},{x3}}
Inequalities
In addition to the equals sign, “ = “ expressions can be related with the symbols: ≠, <, >, ⩽, ⩾ , read “not equal to”, “less than”, “greater then”, “less than or equal to”, “greater than or equal to”. Inequalities are solved using the same strategy as equations: (get the numbers on one side and the variable terms on the other, combine like terms as needed and isolate the variable by dividing both sides by the coefficient of the variable term (or multiplying both sides by its reciprocal)) with one important difference: When multiplying or dividing both sides of an inequality by a negative number you must reverse the direction of the inequality symbol. In Mathematica the built-in function Reduce can be used to solve inequalities.
In addition to the equals sign, “ = “ expressions can be related with the symbols: ≠, <, >, ⩽, ⩾ , read “not equal to”, “less than”, “greater then”, “less than or equal to”, “greater than or equal to”. Inequalities are solved using the same strategy as equations: (get the numbers on one side and the variable terms on the other, combine like terms as needed and isolate the variable by dividing both sides by the coefficient of the variable term (or multiplying both sides by its reciprocal)) with one important difference: When multiplying or dividing both sides of an inequality by a negative number you must reverse the direction of the inequality symbol. In Mathematica the built-in function Reduce can be used to solve inequalities.
In[]:=
Reduce[9v+2(-9v-8)>=8v+7+9,v]
Out[]=
v≤-
32
17
The built-in function NumberLinePlot can be used to plot the solution set to the inequality provided a domain is given
The built-in function NumberLinePlot can be used to plot the solution set to the inequality provided a domain is given
In[]:=
NumberLinePlot[%,{v,-20,20}]
Out[]=
Absolute Value Inequalities
Absolute value inequalities have a standard form: |x - a| < b which is read: “the absolute value of x minus a is less than b” . Solutions to this inequality describe the following set of numbers: “all the numbers within b units of a” It is solved by drawing a number line, plotting the three: points a - b, a and a + b . If the inequality sign is less than or less than or equal to shade the numbers between a - b and a + band write the inequality a - b < x < a + b for the solution If its greater than or greater than or equal to the solution set is the numbers less than a - b or greater than a + b, (the numbers beyond b units from a) and write x < a - b or x > a + b
Absolute value inequalities have a standard form: |x - a| < b which is read: “the absolute value of x minus a is less than b” . Solutions to this inequality describe the following set of numbers: “all the numbers within b units of a” It is solved by drawing a number line, plotting the three: points a - b, a and a + b . If the inequality sign is less than or less than or equal to shade the numbers between a - b and a + band write the inequality a - b < x < a + b for the solution If its greater than or greater than or equal to the solution set is the numbers less than a - b or greater than a + b, (the numbers beyond b units from a) and write x < a - b or x > a + b
In[]:=
Reduce[Abs[v-7]>2,v,Reals]
v<5||v>9(*thisisread"v less than 5 or v greater than 9".Thesoltionisplottedbelow.*)
In[]:=
NumberLinePlot[%,{v,-20,20}]
Out[]=
In[]:=
Reduce[Abs[x-3]<4,x,Reals]
Out[]=
-1<x<7
(*Thisisread" x is greater than -1 and less than 7".Alternately,allthenumbersbetween-1andpositive7.Thesolutionisplottedbelow*)
In[]:=
NumberLinePlot[%,{x,-20,20}]
Out[]=
Numbers, Operations, Properties