Special Cases in Exam Review

1. Exam Review

2. Special Cases This is a special case of the standard form where A = 0 and B = 1, or of the slope-intercept form where the slope M = 0. The graph is a horizontal line with y-intercept equal to b. There is no x-intercept, unless b = 0, in which case the graph of the line is the x-axis, and so every real number is an x-intercept. This is a special case of the standard form where A = 1 and B = 0. The graph is a vertical line with x-intercept equal to a. The slope is undefined. There is no y-intercept, unless a = 0, in which case the graph of the line is the y-axis, and so every real number is a y-intercept. In this case all variables and constants have canceled out, leaving a trivially true statement. The original equation, therefore, would be called an identity and one would not normally consider its graph (it would be the entire xy-plane). An example is 2x + 4y = 2(x + 2y). The two expressions on either side of the equal sign are always equal, no matter what values are used for x and y. In situations where algebraic manipulation leads to a statement such as 1 = 0, then the original equation is called inconsistent, meaning it is untrue for any values of x and y An example would be 3x + 2 = 3x − 5.

3. Equations containing fractional coefficients • y(2y - 1) / 2 = 1 - y / 3 • Any time you are solving an equation that has fractions in it, the best approach is to clear the fractions first.To clear fractions, you need to identify the LCD (lowest common denominator). In this case, with a 2 and a 3 as your denominators, the LCD would be 6. Now, we just need to multiply each term by 6. For instance, if you have y/3 and you multiply by 6, you get 6y/3 which now simply reduces to 2y. Again, it's important you multiply _all_ terms by 6.Once that is done and everything is reduced, you should not have any denominators left. From that point, you'll need to distribute, simplify, and get the equation equal to zero. Then you can solve it either by factoring (if possible) or by using the quadratic formula.

4. Equations with variables in the denominator. • 5/2w - 1/3 = 5/6w - 1/8you can get rid of the fractions by mulitplying through the equation with the common deniminator, which is 24w24w(5/2w) - 24w(1/3) = 24w(5/6w) - 24w(1/8)you get:12(5) - 8w(1) = 4(5) - 3w(1)Simplified:60 - 8w = 20 - 3wAdd 8w to both sides to take it to the right side:60 - 8w + 8w = 20 - 3w + 8w60 = 20 + 5wNow subtract 20 from both sides to take it to the left side60 - 20 = 20 - 20 + 5w30 = 5wNow divide both sides by 5 to solve for w30/5 = 5/5 w6 = wSolution: w = 6

5. Addition property of equality • If the same number is added to both sides of an equation, the two sides remain equal. That is, • if x = y, then x + z = y + z. • 3 + 2 = 3 + 25 = 5. It's true.

6. Multiplication property of equality • The two sides of an equation remain equal if they are multiplied by the same number. That is: for any real numbers a, b, and c, if a = b, then ac = bc.

7. Reflexive Property • The reflexive property of equality says that anything is equal to itself. • A=A • 4=4

8. Symmetric Property of Equality • symmetric property, If A = B, then B = A • 7=5 • 5=7

9. Transitive property of Equality • The transitive property of equality states for any real numbers a, b, and c: • If a = b and b = c, then a = c. • For example, 5 = 3 + 2. 3 + 2 = 1 + 4. So, 5 = 1 + 4. • Another example: a = 3. 3 = b. So, a = b.

10. 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: (a + b) + c = a + (b + c) • 2 + 5) + 4 = 11 or 2 + (5 + 4) = 11 • (2 + 3) + 4 = 2 + (3 + 4)

11. 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:(a.b) .c = a. (b.c) • 2(3×4) = (2×3)4. • (2 * 3) * 4 = 2 * (3 * 4)

12. Commutative property of addition • The property of addition that allows two or more addends to be added in any order without changing the sum;a + b = b + a • c + 4 = 4 + c • (2 + 5) + 4r = 4r + (2 + 5)

13. Commutative property of multiplication • When two numbers are multiplied together, the product is the same regardless of the order of the multiplicands. • 4 * 2 = 2 * 4 • 4 × 7 = 7 × 4. Whether you multiply 4 by 7 or 7 by 4, the product is the same, i.e. 28.

14. Distributive property • The distributive property is actually a very simply concept to learn and apply. It will allow you to simplify something like 3(6x + 4), where you have a number being multiplied by a set of parenthesis • 6(4 + 2) = 24+12=36

15. Property of Opposites or Inverse. • When you add a number to its opposite you get zero a+(-a)=0 • 6 + -6 = 0 • 30 + - 30 = 0

16. Property of Reciprocals • A reciprocal is the number you have to multiply a given number by to get 1. • you have to multiply 2 by 1/2 to get 1. therefore the reciprocal of 2 is 1/2 • When you are dividing fractions, such as 6/3 divided by 4/3, then you can multiply the first fraction by the inverse of the first. Therefore, it becomes 6/3 multiplied by 3/4.

17. 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. For example, 4 + 0 = 4, - 11 + 0 = - 11, y + 0 = y are few examples illustrating the identity property of addition. • 5 + 0 = 5.

18. Identity property of multiplication • The identity property of multiplication, also called the multiplication property of one says that a number does not change when that number is multiplied by 1. • 3 × 1 = 3 • 10 × 1 = 10 • 6 × 1 = 6 • 68 × 1 = 68

19. Multiplicative property of zero • The product of 0 and any number results in 0.That is, for any real number a, a × 0 = 0. • 6 * 0=0 • 9 *0=0 • 100000000 * 0 = 0

20. Closure property of addition • The closure property of addition says that if you add together any two numbers from a set, you will get another number from the same set. If the sum is not a number in the set, then the set is not closed under addition. • 3 + 6 = 9 • 1.5 + 7.2 = 8.7

21. Closure Property of Multiplication • Take any two real numbers. Multiply them.  The product that you get is another real number. This is always true. So we can say that the real numbers are closed under multiplication. • 5 × 8 = 40 • 3.4 × 5 = 17.0

22. Product of Powers property • This property states that to multiply powers having the same base, add the exponents. • 22 × 25 = 4 × 32 = 128 is the same as 22+5 = 27 = 128.

23. Power of a Product Power • This property states that the power of a product can be obtained by finding the powers of each factor and multiplying them. • (3 × 4)2 = 122 = 144 is the same as 32 × 42 = 9 × 16 = 144.

24. Power of Power Property • This property states that the power of a power can be found by multiplying the exponents. • (22)3 = 43 = 64  • 22×3 = • 26 = 64.

25. Quotient of Powers Property • This property states that to divide powers having the same base, subtract the exponents. •  54 /53 is the same as 54-3 = 51 = 5. • \

26. Power of 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. • is the same as

27. Zero Power Property • Zero - Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. • That is, if XY = 0, then X = 0 or Y = 0 or both X and Y are 0. • x2 – 4x = 0         x (4 – x) = 0                                     x (4 – x) = 0 • they all equal zero

28. Negative Power Property • When you have a negative exponent on, say, 4, it will be written 4^-2You basically take the reciprocal of it and change the exponent to a positive one. 4^-2 would be 1/4^2 • 5-2 × 52 = 5(-2 + 2) = 50We know 52 = 25, and we know 50 = 1. So, this says that 5-2 × 25 = 1. What number times 25 equals 1? That would be its multiplicative inverse, 1/25.

29. 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. • Suppose you want to solve the equation x2 + x – 20 = 0.You can factor the left side as:(x + 5)(x – 4) = 0Now, by the zero product property, eitherx + 5 = 0   or   x – 4 = 0,

30. Product of Roots Propert • states that for two numbers a and b ≥ 0, √ab = √a · √b. • √45√45 = √3 · 3 · 5 = √32 · √5= 3√5

31. Quotient of Roots Property • tates that for any numbers a and b, where a ≥ 0 and_ b≥0, _a=√a.√b√b

32. Solving First power iniqualities with one variable. (One sign) • Add up all the numbers on the left side of the inequality.Step 2 • Add up all the numbers on the right side of the inequality.Step 3 • Add up the variable with coefficients (i.e. 3x+4x) on the left side of the inequality.Step 4 • Add up the variable with coefficients (i.e. 2x+x) on the right side of the inequality. • Subract the number on the left side (if it is a positive number) from both sides of the inequality or add the number on the left side (if it is a negative number) from both sides of the inequality. • Subract the variable with a coefficient on the right side of the inequality (if it is a positive variable with a coefficient ) from both sides of the inequality or add the variable with a coefficient on the right side of the inequality (if it is a negative variable with a coefficient) from both sides of the inequality. • Simplify (if needed) by dividing (if the coefficient is an integer) both sides of the inequality by the coefficient (i.e. the 8 in 8x) or multiplying both sides of the inequality by the reciprocal of the coefficient (if the coefficient is a fraction). Note: The inequality sign is reversed if both sides of the inequality are multiplied or divided by a negative number.

33. Linear equations in two variables • In general, a solution of a system in two variables is an ordered pair that makes BOTH equations true. In other words, it is where the two graphs intersect, what they have in common.  So if an ordered pair is a solution to one equation, but not the other, then it is NOT a solution to the system. A consistent system is a system that has at least one solution.An inconsistent system is a system that has no solution. 
 The equations of a system are dependent if ALL the solutions of one equation are also solutions of the other equation.  In other words, they end up being the same line.The equations of a system are independent if they do not share ALL solutions.  They can have one point in common, just not all of them.

34. Linear systems (substitution) • The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you back-solve for the first variable • 4x + y = 24
y = –4x + 24 • 2x – 3(–4x + 24) = –2
2x + 12x – 72 = –2
14x = 70
x = 5

35. Linear Systems (Addition) • The addition method allows you to add the equations given to you in a system. • The addition method says we can just add everything on the left hand side and add everything on the right side and keep the equal sign in between. • Now it is possible to solve the new equation and get x=4. once you know one of the variables substitute it into either equation to find the other variable in this case y = 2

36. Linear systems (terms) • A system of linear equations either has no solutions, a unique solution, or an infinite number of solutions. If it has solutions it is said to be consistent, otherwise it is inconsistent. A system of linear equations in which there are fewer equations than unknowns is said to be underdetermined. These are the systems that often give infinitely many solutions. A system of equations in which the number of equations exceeds the number of unknowns is said to be overdetermined. In an overdetermined system, anything can happen, but such a system will often be inconsistent.

37. Factoring • Factor GCF  for any # terms • Difference of Squares  binomials • Sum or Difference of Cubes  binomials • PST (Perfect Square Trinomial)  trinomials • Reverse of FOIL  trinomials • Factor by Grouping  usually for 4 or more terms

38. Examples • GCF: 5x3 – 10x2 – 5x • 5x(x2 – 2x – 1) • Differnce of Sqaures:75x4 – 108y2 • GCF first! 3(25x4 – 36y2) • Sum or difference of cubes: a3 - b3 • (a - b) (a2 + ab + b2) • Difference of Cubes: m6 – 125n3 • \ • (m2 – 5n) (m4 + 5m2n + 25n2) • PST: 9x2 – 30x + 25 • (3x – 5) 2 • Reverse Foil: 6x2 – 17x + 12 • (3x – 4)(2x – 3) • Facrtoring by grouping: x2 + 6x + 9 – 4y2 [x2 + 6x + 9 ]– [4y2] (x + 3) 2 – 4y2 [(x + 3) + 2y] [(x + 3) – 2y]

39. Rational Expressions • To simplify a rational expression, we first factor both the numerator and denominator completely then reduce the expression by cancelling common factors. • 4x – 2 /2x – 1 • 2(2x - 1) = 2 • 1(2x – 1) • Addition and subtraction are the hardest things you'll be doing with rational expressions because, just like with regular fractions, you'll have to convert to common denominators. Everything you hated about adding fractions, you're going to hate worse with rational expressions. But stick with me; you can get through this! • find the common denominator, I first need to find the least common multiple (LCM of the comin denominator. • Both the numerators and the denominators multiply together Common factors may be cancelled before multiplying

40. Quadratic Equations (Factoring) • Well, suppose you have a quadratic equation that can be factored, like • x2+5x+6=0. • This can be factored into • (x+2)(x+3)=0. • So the solutions must be x=-2 and x=-3. • Note that if your quadratic equation cannot be factored, then this method will not work

41. Quadratic equations (square root) • (x – 5)2 – 100 = 0 (x – 5)2 = 100 x – 5 = ±10 x = 5 ± 10 x = 5 – 10  or  x = 5 + 10 x = –5   or  x = 15

42. Quadratic Formuala • The Quadratic Formula uses the "a", "b", and "c" from "ax2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients". The Formula is derived from the process of completing the square, and is formally stated as: •  For ax2 + bx + c = 0, the value of x is given by:

43. Quadratic equation (Discriminant) • The discriminant is a number that can be calculated from any quadratic equation A quadratic equation is an equation that can be written as • ax ² + bx + c where a ≠ 0 • The discriminant in a quadratic equation is found by the following formula and the discriminant provides critical information regarding the nature of the roots/solutions of any quadratic equation.

44. Functions F(x) • The same is true of "y" and "f(x)" For functions, the two notations mean the exact same thing, but "f(x)" gives you more flexibility and more information. You used to say "y = 2x + 3; solve for y when x = –1". Now you say "f(x) = 2x + 3; find f(–1)" (pronounced as "f-of-x is 2x plus three; find f-of-negative-one"). You do exactly the same thing in either case: you plug in –1 for x, multiply by 2, and then add the 3, simplifying to get a final value of +1.

45. Functions (Domian and Range) • Definition of the Domain of a Function For a function f defined by an expression with variable x, the implied domain of f is the set of all real numbers variable x can take such that the expression defining the function is real. The domain can also be given explicitly • Definition of the Range of a Function The range of f is the set of all values that the function takes when x takes values in the domain.

46. Linear Functions • Linear functions are functions that have x as the input variable, and x is raised only to the first power. Such functions look like the ones in the above graphic. Notice that x is raised to the power of 1 in each equation. • y = mx + b • y = m(x - x 1 ) + y 1      or      y - y 1 = m(x - x 1 ) • Ax + By + C = 0     or     y = (-A/B)x + (-C/B)

47. Parabola • Determine whether the parabola opens upward or downward. • a.    If a > 0, it opens upward. • b.    If a < 0, it opens downward. • 2.    Determine the vertex. • a.    The x-coordinate is . • b.    The y-coordinate is found by substituting the x-coordinate, from          Step 2a, in the equation y = ax2 + bx + c. • 3.    Determine the y-intercept by setting x = 0. • 4.    Determine the x-intercepts (if any) by setting y = 0, i.e., solving the equation        ax2 + bx + c = 0. • 5.    Determine two or three other points if there are no x-intercepts.

48. Simplifying expressions with exponents • The rules tell me to add the exponents. But I when I started algebra, I had trouble keeping the rules straight, so I just thought about what exponents mean. The " x6 " means "six copies of x multiplied together", and the " x5 " means "five copies of x multiplied together". So if I multiply those two expressions together, I will get eleven copies of x multiplied together. That is: • x6 × x5 = (x6)(x5)              = (xxxxxx)(xxxxx)    (6 times, and then 5 times)             = xxxxxxxxxxx         (11 times)              = x11  

49. Simplifying expressions with radicals • Simplify terms with Like Radicals by combining these Terms. • 2) Simplify radicals by extracting perfect powers from the radicand to Reduce the radical. • 3) Rationalize fractions with radicals by clearing radicals from the denominator. • The two terms here are Like Terms with a common radical factor. Since they are like terms, you can combine them.   When we combine the numerical coefficients of each term, 2 + -1 = 1, we get the following results.

50. Word problems • There are b boys in the class.  This is three more than four times the number of girls.  How many girls are in the class? • Solution.   Again, let x represent the unknown number that you are asked to find:  Let x be the number of girls. • (Although b is not known, it is not what you are asked to find.) • The problem states that "This" -- b -- is three more than four times x: •  4x + 3=b.  Therefore, 4x=b − 3  x=b − 3   4.