ALGEBRA 1 REVIEW Tony Hren Algebra 1 Review May 14, 2010

Addition Property (of Equality) If the same number is added to both sides of an equation, the two sides remain equal. Example: If a=b, then a+c = b+c http://www.icoachmath.com/sitemap/Addition_Property_of_Equality_and_Inequality.html Multiplication Property (of Equality) You can multiply both sides of an equation by the same nonzero number, and this won't change the truth of the equation. Example: If a=b, then ac=bc http://www.onemathematicalcat.org/algebra_book/online_problems/mult_prop_eq.htm

Reflexive Property (of Equality) Example: a=a Symmetric Property (of Equality) Example: If a=b, then b=a Transitive Property (of Equality) Example: If a=b, and b=c, then a=c

Associative Property of Addition The property which states that for all real numbers a, b, and c, their sum is always the same, regardless of their grouping. Example: (a + b) + c = a + (b + c) http://www.harcourtschool.com/glossary/math_advantage/definitions/associative_add7.html Associative Property of Multiplication The property which states that for all real numbers a, b, and c, their product is always the same, regardless of their grouping. Example: (a x b) x c = a x (b x c) http://www.harcourtschool.com/glossary/math_advantage/definitions/associative_mul7.html

Commutative Property of Addition The Commutative Property of Addition states that changing the order of addends does not change the sum Example: if a and b are two real numbers, then a + b = b + a. http://www.icoachmath.com/SiteMap/CommutativePropertyofAddition.html Commutative Property of Multiplication The Commutative Property of Multiplication states that changing the order of the factors does not change the product. Example: if a and b are two real numbers, then a × b = b × a.

Distributive Property (of Multiplication over Addition) The property which states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. Example: a(b + c) = a X b + a X c http://www.harcourtschool.com/glossary/math_advantage/definitions/distributive_p7.html

Property of Opposites or Inverse Property of Addition The property that states the sum of a number and its opposite is always zero. Example: a+(-a)=0 http://www.washoe.k12.nv.us/ecollab/washoemath/dictionary/vmd/full/a/dditionpropertyofopposites.htm Prop of Reciprocals or Inverse Prop. of Multiplication For every non-zero real number a there is a unique real number 1/a such that: Example: a( )=1 http://everyonehatesmath.com/property-of-reciprocals/

Identity Property of Addition Identity property of addition states that the sum of zero and any number or variable is the number or variable itself. Example: a+0=a Identity Property of Multiplication http://www.northstarmath.com/Sitemap/IdentityPropertiesofAdditionandMultiplication.html Identity property of multiplication states that the product of 1 and any number or variable is the number or variable itself. Example: 4 x 1=4

Multiplicative Property of Zero Any number multiplied by zero equals zero. Example: a(0)=0 Closure Property of Addition Closure property of addition states that the sum of any two real numbers equals another real number. Example: If a and b are real numbers, then a + b equals a real number. Closure Property of Multiplication Closure property of multiplication states that the product of any two real numbers equals another real number. Example: If a and b are real numbers, then a x b is equal to a real number.

Product of Powers Property This property states that to multiply powers having the same base, add the exponents. Example: for a real number non-zero a and two integers m and n, am × an = am+n. Power of a Product Property http://www.icoachmath.com/sitemap/Power_Properties.html This property states that a product of a power can be obtained by finding the powers of each property and multiplying them. Example: (ab)m = am × bm Power of a Power Property This property states that the power of a power can be found by multiplying the exponents. Example: (am)n = amn

Quotient of Powers Property This property states that to divide powers having the same base, subtract the exponents. Example:           . http://www.icoachmath.com/sitemap/Power_Properties.html Power of a Quotient Property This property states that the power of a quotient can be obtained by finding the powers of numerator and denominator and dividing them. Example:

Negative Power Property                                   Zero Power Property When a number is raised to the zero power, it is always equal to 1. Example: a0=1 Negative Power Property To solve for negative exponents, write the reciprocal of the expression, and change the negative to the positive power. Example: http://www.mathsisfun.com/algebra/negative-exponents.html

Zero Product Property The Zero Product Property simply states that if ab = 0, then either a = 0 or b = 0 (or both). A product of factors is zero if and only if one or more of the factors is zero. Example: ab=0. If this is true, a, b, or a and be must be equal to zero.

Product of Roots Property The factors of the square root of a number are equal to the square root of one factor of the original number multiplied by the square root of another. = X Example: Quotient of Roots Property The roots of a square root of a quotient are equal to the square root of the numerator written over/divided by the square root of the denominator. http://www.tutorvista.com/math/square-root-property-calculator Example: =

Root of a Power Property Example: a4 = a2 x a2 = (a)(a)(a)(a) http://hotmath.com/hotmath_help/topics/properties-of-exponents.html Power of a Root Property Example:

PROPERTY QUIZ!

Click when you’re ready for the answers. 1.) (am)n = amn 8.) (a x b) x c = a x (b x c) Power of a Power Property Associative Poperty of Multiplication 2.) If a=b, then b=a 9.) If a and b are two real numbers, then a × b = b × a. Symmetric Property (of Equality) 3.) If a=b, then a+c = b+c Commutative Property of Multiplication 10.) If a and b are two real numbers, then a + b = b + a. Addition Property (of Equality) 4.) a+(-a)=0 Property of Opposites/Inverse Operation of Addition Commutative Property of Addition 5.) If a=b, then ac=bc 11.) a(b + c) = a X b + a X c Multiplication Property of Equality Dist. Prop. (of Multiplication over Division) 6.) If a=b, then ac=bc 12.) a0=1 Multiplicative Property of Zero 7.) If a=b, and b=c, then a=c Zero Power Property Transitive Property of Equality

Solving 1st Power Inequalities in One Variable -With only one inequality sign Example: 6x > 24 A linear equation has only one value for the solution that holds true. For example, the linear equation 6x = 24 is a true statement only when x = 4. However, the linear inequality 6x > 24 is satisfied when x > 4. So, there are many values of x which will satisfy the inequality 6x > 24, which is the same thing as x > 4, which is the answer. As shown below, x > 4. This is the answer because after dividing each side by 6, we are left with x > 4. http://www.mathsteacher.com.au/year10/ch02_linear_equations/07_subtract/solve.htm

Another Example The solution set consists of all numbers less than or equal to –2, as shown on the following number line. http://www.mathsteacher.com.au/year10/ch02_linear_equations/07_subtract/solve.htm

Special Case: Division and Multiplication of Negative Numbers With only one inequality sign if an inequality is multiplied (or divided) by the same negative number, then: Example: -3x > 27 [Divide each side by (-3)] Answer: x < -9 http://www.mathsteacher.com.au/year10/ch02_linear_equations/07_subtract/solve.htm

Conjunctions Examples: Must satisfy both conditions of the inequality RESULT: Examples: Graph A: 4 ≤ x < 9 Graph B: 4 < x ≤ 9 Graph C: 4 < x < 9 Graph D: 4 ≤ x ≤ 9 http://image.tutorvista.com/Qimages/QD/28881.gif

Disjunctions Example: x ≤ -3 OR x > 2 Must satisfy either one or both of the conditions < AND > have open endpoints, while ≥ and ≤ have closed endpoints Example: x ≤ -3 OR x > 2 http://hotmath.com/images/gt/lessons/genericalg1/f-352-21-ex-1.gif

Special Cases If there are no solutions that work, the answer is Ø If every number works, the answer is {reals} If there is a disjunction in the same direction, only one arrow is needed

Linear Equations with Two Variables Slopes and Equations of Lines Positive: Lines that rise as you move from left to right. Negative: Lines that fall as you move from left to right. Rising and Falling lines are associated with rise/run (rise over run) which is the same as the difference between y-coordinates/difference between x-coordinates. Horizontal Lines: Each of these has a slope of 0, and refers to a straight line from left to right. Vertical Lines: Each of these has no slope, and refers to a straight line running up and down. http://www.google.com/imgres?imgurl=http://www.learningwave.com/lwonline/algebra_section2/graphics/typesslope2.gif&imgrefurl=http://www.learningwave.com/lwonline/algebra_section2/slope2.html&usg=__fF7XXBj0ZQWoDV0dCqXQObS45x8=&h=192&w=203&sz=6&hl=en&start=5&um=1&itbs=1&tbnid=NLy_ChIj2-lDfM:&tbnh=99&tbnw=105&prev=/images%3Fq%3Dthe%2B4%2Btypes%2Bof%2Bslopes%26um%3D1%26hl%3Den%26sa%3DN%26ndsp%3D20%26tbs%3Disch:1 Positive Negative Vertical Horizontal

Point-Slope formula: How to Graph and find Intercepts? In the slope-intercept formula, the equation is y=mx+b., whereas ‘m’ is equal to the slope, ‘b’ is the y-intercept and the ‘y’ and ‘x’ are the coordinates. Start graphing by placing a point on the y-intercept. Then follow the slope to make your line. The x-intercept is where on the x-axis the line passes through. http://www.saskschools.ca/curr_content/matha30rev1/lesson3-4/ponitslopeformula.jpg Standard Form Ax + By = C A, B, C are integers (positive or negative whole numbers) No fractions nor decimals in standard form. Traditionally the "Ax" term is positive. http://www.algebralab.org/studyaids/studyaid.aspx?file=algebra1_5-5.xml

Linear Systems Substitution Method: First solve one equation for one of the variables. Substitute this expression in the other equation and solve for the other variable next. Then, substitute this value in the equation of the first step and solve. Example: z=4y-4

Choices: dependent, inconsistent, consistent Addition/Subtraction Method (Elimination Method): First add the similar terms of the two equations to find the x. Then solve the resulting equation. Substitute that answer for the other variable to find y. Then check your final answers in both equations. Terms Quiz Choices: dependent, inconsistent, consistent 1.) A system in whichthe solution id all points on the line dependent 2.) A system in which the lines cross on one point. consistent 3.) A system which is false or null set because it is parallel inconsistent

Factoring GCF: For any number of terms, factor first the common factors of the expressions, and then fill in what is left over. Difference of Squares: For binomials, first find the GCF, then find the squares and simplify what is left over, hopefully finding conjugates. Sum or Difference of Cubes: For binomials, simply find the square or cube root from each side. PST: For trinomials, If 1st & 3rd terms are squares and the middle term is twice the product of their square roots, use this method by reverse foil. Reverse Foil: for trinomials, do the opposite of F O+I L. Factor By Grouping: Usually do this with 4 or more terms, and take the roots of the algebraic expressions.

Rational Expressions To simplify a rational expression: Completely factor numerators and denominators. Reduce common factors. Example : Simplify Examples: Addition: 3/5 + 1/5 = 4/5. Subtraction: 11/12 – 4/12 = 7/12. Division: ¾ / 2 = ¾ x ½ =3/8. Multiplication: ¾ x ¾ =9/4 = 2 and ¼. http://www.cliffsnotes.com/study_guide/Simplifying-Rational-Expressions.topicArticleId-38949,articleId-38901.html

Quadratic Equations in one Variable Factoring: x2+5x+6=0 = (x+2)(x+3)=0 (-2, -3) Taking Square Root of each side: x2= 4 x=2. Quadratic Formula Completing the Square: (x – 4)2 = 5 x – 4 = ± sqrt(5) x = 4 ± sqrt(5) x = 4 – sqrt(5)  and  x = 4 + sqrt(5) http://www.sosmath.com/algebra/quadraticeq/quadraformula/quadraformula.html http://www.purplemath.com/modules/sqrquad.htm The discriminant tells you how many X-axis intercepts a polynomial function has.

Functions F(x) means the same thing as “y” but gives more information. The expression "f(x)" means "plug a value for x into a formula f "; You solve it the same way you would for a “y”. Not all relations are function. y = √(x + 4) The domain of the function is x ≥ −4, because x cannot take values less than -4. the range for this function is y ≥ 0, because There is no value of x that we can find such that we will get a negative value of y. In order to find a linear equation when given two pairs of data, follow the rule: y2-y1/x2-x1. For example if you had the ordered pairs (2,3) and (1,5), you would first do 5-3, which is equal to 2, over 1-2, which is equal to -1, which is the slope.  Then substitute zero for one of the values and solve for x and y. http://www.intmath.com/Functions-and-graphs/2a_Domain-and-range.php

Parabolas WRITE YOUR EQUATION ON PAPER. IF NECESSARY, TRY TO REARRANGE THE EQUATION INTO THE FORM OF A PARABOLA, Y - K = A (X - H)^2. (OUR EXAMPLE IS Y - 3 = - 1/6 (X + 6)^2, WHERE ^ DENOTES AN EXPONENT.) FIND THE VERTEX OF THE PARABOLA. THE VERTEX IS THE EXACT CENTER OF THE PARABOLA, THE KEY COMPONENT. USING THE FORMULA FOR A PARABOLA, Y - K = A (X - H)^2, THE VERTEX X-COORDINATE (HORIZONTAL) IS "H" AND THE Y-COORDINATE (VERTICAL) IS "K." FIND THESE TWO VALUES IN YOUR ACTUAL EQUATION. (OUR EXAMPLE IS H = -6 AND K = 3.) FIND THE Y-INTERCEPT BY SOLVING THE EQUATION FOR "Y." SET "X" TO "0" AND SOLVE FOR "Y." (OUR EXAMPLE IS Y = -3.) FIND THE X-INTERCEPT BY SOLVING THE EQUATION FOR "X." SET "Y" TO "0" AND SOLVE FOR "X." WHEN TAKING THE SQUARE ROOT OF BOTH SIDES, THE SINGLE NUMBER SIDE OF THE EQUATION BECOMES BOTH POSITIVE AND NEGATIVE (+/-), RESULTING IN TWO SEPARATE SOLUTIONS, ONE USING THE POSITIVE AND ONE USING THE NEGATIVE. DRAW A BLANK LINE GRAPH ON GRAPH PAPER. DETERMINE THE SIZE ANDAREA OF THE GRAPH. A PARABOLA GOES TO INFINITY, SO THE GRAPH IS ONLY A SMALL PORTION NEAR THE VERTEX, WHICH IS THE TOP OR BOTTOM OF THE PARABOLA. THE GRAPH NEEDS TO BE DRAWN IN PROXIMITY TO THE VERTEX. THE X AND Y-INTERCEPTS TELL THE ACTUAL POINTS THAT APPEAR ON THE GRAPH. DRAW A STRAIGHT HORIZONTAL LINE AND A STRAIGHT VERTICAL LINE INTERCEPTING AND PASSING THROUGH THE HORIZONTAL LINE. DRAW AN ARROW AT BOTH ENDS OF BOTH LINES TO REPRESENT INFINITY. MARK SMALL TICK LINES ON EACH LINE AT EQUAL INTERVALS REPRESENTING NUMERAL INCREMENTS IN THE VICINITY OF THE SIZE OF THE COORDINATES. MAKE THE GRAPH A FEW TICKS LARGER THAN THESE COORDINATES. PLOT THE PARABOLA ON THE LINE GRAPH. PLOT THE VERTEX, X-INTERCEPT, AND Y-INTERCEPTS POINTS ON THE GRAPH WITH LARGE DOTS. CONNECT THE DOTS WITH ONE CONTINUOUS U-SHAPED LINE AND CONTINUE THE LINES TO NEAR THE END OF THE GRAPH. DRAW AN ARROW AT BOTH ENDS OF THE PARABOLA LINE TO REPRESENT INFINITY. http://www.ehow.com/how_4546044_graph-parabola.html

Simplifying Expressions with Exponents and Radicals Simplify (–46x2y3z)0  This is simple enough: anything to the zero power is just 1. (-46x2y3z)0 =1 x6 × x5 = (x6)(x5)               = (xxxxxx)(xxxxx)    (6 times, and then 5 times)              = xxxxxxxxxxx         (11 times)               = x11   = Radical Expressions Examples http://www.algebralab.org/lessons/lesson.aspx?file=algebra_radical_simplify.xml

Word Problems Many single-variable algebra word problems have to do with the relations between different people's ages. For example: Al's father is 45. He is 15 years older than twice Al's age. How old is Al?We can begin by assigning a variable to what we're asked to find. Here this is Al's age, so let Al's age = x. We also know from the information given in the problem that 45 is 15 more than twice Al's age. How can we translate this from words into mathematical symbols? What is twice Al's age? Al's age is x, so twice Al's age is 2x, and 15 more than twice Al's age is 15 + 2x. That equals 45. Now we have an equation in terms of one variable that we can solve for x: 45 = 15 + 2x. original statement of the problem:  45 = 15 + 2x subtract 15 from each side: 30 = 2x divide both sides by 2: 15 = x Since x is Al's age and x = 15, this means that Al is 15 years old

Word Problems (cont.) Ned got a 12% discount when he bought his new jacket. If the original price, before the discount, was $50, how much was the discount? Word problems tend to be even wordier than this one. The solution process involves making the problem simpler and simpler, until it's a math problem with no words. Step 1. Identify what they're asking for, and call it x. x = amount of the discount. Step 2. Use the information given to write an equation that relates the quantities involved. 12% of 50 dollars = the amount of the discount (x). Step 3. Translate into Math: (12/100) * 50 = x. Step 4. Solve for x: 6 = x. This means that Ned's 12% off amounted to a $6 discount. http://www.algebra.com/algebra/homework/Percentage-and-ratio-word-problems/Percentage-Word-Problems-(discount).lesson

Word Problems (cont.) Step 2: Fill in the table with information given in the question. John has 20 ounces of a 20% of salt solution. How much salt should he add to make it a 25% solution? The salt added is 100% salt, which is 1 in decimal. Change all the percent to decimals Let x = amount of salt added. The result would be 20 + x. Adding To The Solution Mixture Problems: Example 1: John has 20 ounces of a 20% of salt solution, How much salt should he add to make it a 25% solution? Solution: Step 1: Set up a table for salt.   original added result concentration amount   original added result concentration 0.2 1 0.25 amount 20 x 20 + x Step 3: Multiply down each column.   original added result concentration 0.2 1 0.25 amount 20 x 20 + x multiply 0.2 × 20 1 × x 0.25(20 + x) http://www.onlinemathlearning.com/mixture-problems.html#add Step 4:original + added = result 0.2 × 20 + 1 × x = 0.25(20 + x) 4 + x = 5 + 0.25x Isolate variable x x – 0.25x = 5 – 4 0.75x = 1 Answer: He should add      ounces of salt.

Word Problems (Cont.) You put $1000 into an investment yielding 6% annual interest; you left the money in for two years. How much interest do you get at the end of those two years? In this case, P = $1000, r = 0.06 (because I have to convert the percent to decimal form), and the time is t = 2. Substituting, I get: I = (1000)(0.06)(2) = 120 I will get $120 in interest. You invested $500 and received $650 after three years. What had been the interest rate? For this exercise, I first need to find the amount of the interest. Since interest is added to the principal, and since P = $500, then I = $650 – 500 = $150. The time is t = 3. Substituting all of these values into the simple-interest formula, I get: 150 = (500)(r)(3) 150 = 1500r 150/1500 = r = 0.10 Of course, I need to remember to convert this decimal to a percentage. I was getting 10% interest http://www.purplemath.com/modules/percntof.htm

Line of Best Fit A line of best fit  is a straight line that best represents the data on a scatter plot.  This line may pass through some of the points, none of the points, or all of the points.  Here's an example. Suppose you want to find out whether more hours spent studying will have an affect on a person's mark. You set up an experiment with some people, recording how many hours they spent studying and then recording what happened to their mark. You can see the data in the table at the right. It's difficult to see any pattern in the table, although it's clear that different things happened to different people. One person studied for 1 hour and had their mark go up 2%, while another person who also studied for 1 hour saw a drop of 1%! A graphing calculator helps because if you enter information correctly, it will draw one for you.

I HOPE YOU ENJOYED THIS BEAUTIFUL ALGEBRA 1 PRESENTATION! The end.