Algebra, Equations, and Inequalities

Solve (infinitely many solutions): 2x-y= 5 2y-4x= -10

-10= -10 (answer has no variable, so there are infinitely many solutions. also, one equation is just a multiple of the other meaning the same thing. (only one line).

Infinitely Many solutions

1. If all variables go away in the process of elimination or substitutions and the numbers equal each other, then there is an infinite number of solutions. Note: Also, if one equation is a multiple of another, you have infinite solutions.

Rules for combining inequalities

1. If we have a = b and b = c, we can combine them to get a = c 2. If r < s and s < t, we can say that r < s < t, and so, r < t 3. We cannot draw any conclusion if the same term is greater than both other terms or less then both other terms (e.g. c < f and d < f) 4. We can add inequalities withe same direction. If a > b and c > d, then (a + c) > (b+d). 5. We cannot add inequalities that do not have the same direction. 6. We cannot subtract inequalities with the same direction. 7 We CAN subtract inequalities with the opposite directions. The resultant inequality follows the direction of the first inequality. 8. There are no rules for multiplication and division of inequalities. 9. Remember that any positive number is bigger than any negative number. 10. Adding a positive number always makes the number bigger (x< x + 2/5) 11. Subtracting a positive or adding a negative always makes a number smaller.

Factoring quadratics things to remember

1. The primary method for factoring quadratics only works if the quadratic coefficient is one (x^2). If it is something other than one, use the other methods of factoring (Difference of two squares, GCF, etc.).

Quadratic Equations Overview

1. Usually have two different solutions. But some have one and some have none 2. Process of solving is different from linear equations. 3. Factor a quadratic to a product of two linear binomials. Make sure things equal 0 3.1. x^2+bx+c= (x-p)(x-q)= 0

FOIl Module things to remember

1. You can distribute multiplication across addition and subtraction. You CANNOT distribute exponents across addition and subtraction: (a+b)^2 IS NOT a^2 + b^2 2. So, (a+b)^2 is (a+b)(a+b). Use foil to solve. 3. The sum of a square always takes the following form (a + b)^2= a^2 + 2ab + b^2 4. The square of a difference always takes the following form: (a-b)^2= a^2 -2ab + b^2

Solve for x and y using the elimination method: 7x+3y= 5 2x-3y= 13

1. add equations together: 9x=18 2. x=2; plug two back in to either equation 3. 2(2) - 3y= 13 4. -3y= 9 5. y= -3 6. y=-3 and x=2 (answer)

Quadratic equation solving process

1. get everything on one side of the equation, set equal to zero 2. Divide by any GCF (if necessary) 3. Factor 4. Use the zero product property to create two linear equations, and solve.

Advanced numerical factoring

1. if you can get to the number you want to factor by subtracting a perfect square from another perfect square, factoring becomes easier. 2. You can also use the difference of two squares to simplify decimal problems.

Three equations with three unknowns steps

1. pick two of the three equations and eliminate one variable. 2. pick another pair of the original equations and eliminate the same variable. 3. Use two-equations-with-two unknown techniques to solve for the remaining variables. 4. Plug answer into any original equation find the value of the third variable.

Absolute value inequalities

1. remember |x - 3| is the distance of x from positive 3. 2. remember that |x + 3| is the distance of x from negative 3. 3. The middle of a region on the number line is the average of the two endpoints.

Equations with two different variables

1. these equations without powers, can be represented by a straight line. 2. Nobody can ask you to solve a single equation with two variable because it would have infinite solutions. 3. However, if have two equation with the same two unknown variables, we can usually solve for those variables.

Algebraic simplification rule

1.) We can combine like terms (same variable part) by addition or subtraction. e.g. 15y-8y= 7y 2.) We can add or subtract only like terms, not terms with different variables or powers 3.) Multiplication is commutative (ab = ba) 4.) When an addition sign appears in front of the parentheses, we can simply remove the parentheses 5.) When a subtraction sign appears in front of the parentheses, we must change every sign in the parentheses.

Simplify: 5xy + 7yx

12xy Note: order of variables does not matter because we are working with multiplication.

Simplify: (x^3- 3(x^2)+ 3x) + (x^3+ 3(x^2)-3x)

2(X^3)

Simplify: dffddf A. 6(x^2)- 8x + 10 - 2(x^2) + 3x - 4 B. 3(y^2) + 7xy + 2(x^2) + y^2- 7xy + 2(x^2)

A. 4(x^2)- 5x+ 6 B. 4(y^2)+ 4x2

Inequalities 2 practice: For any number x, which of the following must be greater than x? I. x+2 II. 2x III. x^2

Answer: I only II does not work. If x is negative, the product with be smaller. III does not work. squaring fractions makes them smaller. squaring 1 and 0 produces equalities, not inequalities.

FOIL method

F= First (P+q)(R+s) O= Outer (P+q)(r+S) I= Inner (p+Q)(R+s) L= Last (p+Q)(r+S)

Factoring Quadratics

Find 2 number for which the sum = the midterm and the product equals the last term

Inequality rules

Inequalities remain the same when... 1. we add or subtract the same thing from both sides 2. we multiply or divide both sides by any positive number. The order of inequalities changes when... 1. we multiply or divide both sides by any negative number (e.g. -x>3 becomes x<-3)

Strategies for solving two equation, two unknowns

S1: Substitution use this method when at least one of the variables has a coefficient of 1 or -1 S2: Elimination (linear combination) Use this. Use this when no coefficients are equal to +/-1. Eliminate one variable completely from the equations by adding them together.

Any even power of x is the square of another power

This means the following x^10 = X^5+5 = (X^5)(X^5) = (X^5)^2

Simplify: Y^3+8(y^2)-5(y^2)+5y

Y^3+3(y^2)-3y

Strange operators strategy

You will be given a symbol that nobody has every used before. Just follow the rules give for that symbol to answer the question.

rational expression (definition)

a ratio, a fraction, of two algebraic expressions. (x-3)/(x+2)

Simplify the following (difference of squares: ((0.999856/0.988)-1)

a. ((1-0.000144/1-0.012)-1) b. ((1+012)(1-0.012)/1-0.012)-1) c. 1.012-1 d. 0.012

**Simplifying complex algebra with substitutions practice: Solve for x: (2x-1)^2 + 5(2x-1) = 24

a. (2x-1)^2 + 5(2x-1) -24 = 0 b. u^2 + 5u -24 = 0 c. (U + 8)(U -3) = 0 d. U = -8 or U = 3 e. -8 = 2x - 1 or 3 = 2x -1 f. -7 = 2x or 4 = 2x g. x = -7/2 or x = 2 (answer)

Simplify: (2x+y)(x+2y)

a. (2x^2)+4xy+ xy+ (2y^2) b. (2x^2)+5xy+(2y^2) --answer

If (4p+3Q-4R)/(P-R)= 19, then Q/(P-R)=?

a. (3Q+4P-4R)/(p-R) b. (3Q/(p-R) + (4P-4R)/(P-R) c....+ 4(P-R)/P-R d....+4=19 e...=15 f. 3Q/(P-R)=15 g. Q/(P-R)= 5 (answer)

Difference of two square factor example(s) a. (9x^2)- 16 b. (25x^2) - (64y^2) c. (x^2)(y^2) - 1

a. (9x^2)- 16= (3x +4)(3x - 4) b. = (5x - 8y)(5x + 8y) c. = (xy + 1)(xy - 1)

Given f(x)= x^2- 2x - 1, find f(x^2+3)

a. (x^2 +3)^2 -2(x^2+3) -1 b. ... c. x^4 +4x^2 +2 (answer)

Simplify: (x^3- 3(x^2)+ 3x) - (x^3+ 3(x^2)-3x)

a. (x^3- 3(x^2)+ 3x) - x^3- 3(x^2)+ 3x b. -6(x^2) + 6x (answer)

Inequality practice: -4 < 5 - 3x ≤ 17

a. -9 < -3x ≤ 12 b. 3 > x ≥ -4 Note: remember that what you do to one of the three expressions, you should do to all of them. When you subtract 5 from the middle expression, subtract five from the two other expressions as well.

For positive numbers a and b, let a @ b = (2a^2) + b. What does (1 @ 2) @ 3 equal

a. ... b. 35 (answer)

Practice problem: For positive numbers p and q, let p @ q = p + (1/q). What does (1 @ 2) @ 3 equal?

a. 1 @ 2= 3/2 b. (3/2) @ 3 = 11/6 answer

Find the prime factorization of 9975 (D of S)

a. 10,000-25 b. 100^2-5^2 c. (100-5)(100+5) d. 95x105 e. 19x5x5x3x7

Simplify: a. (3x)(4x^2) b. (7x(^2)y(^2))(6x(y^3) c. (2xy)(3xz)(4yz)

a. 12x^3 b. (42x^3)(y^5) c. 24(X^2)(y^2)(z^2)

Find the prime factorization of 1599

a. 1599= 1600-1 b. 1600-1= 40^2 - 1^2 c. 40^2 - 1^2= (40+1)(40-1) d. (40+1)(40-1)= 41x39 e. 41x39= 41x3x13

Inequality practice: 2/x ≥ 1/3

a. 2 ≥ x/3 b. 6 ≥ x c. 0 < X ≤ 6 Note we know that x can't be 0 because 0 doesn't work in the denominator. It also can't be a negative number because 2/x is greater than the positive fraction 1/3.

If (x^2+12x-45)/x+15 = 22, then x=?

a. 22 =((x+15)(x-3))/x+15 b. 22 =x-3 c. x=25 (answer)

***Find the prime factorization of 2491 (diff. of squares)

a. 2500-9 b. 50^2 - 3^2 c. (50+3)(50-3) d. 53x47

Given the function: f(x)= x^2 + kx +4 find the value of k if f(2) =18

a. 2^2 + 2k + 4 =18 b. 4 +2k +4= 18 c. 2k+8= 18 d. 2k = 10 e. k = 5 (answer).

Absolute Values Solve: |2x+3|= 5

a. 2x+3=5 b. x= 1 OR c. 2x+3= -5 d. 2x=-8 e. x= -4 f. x= 1 or -4 (answer)

Factoring GCF from a trinomial examples a. 2x^4 + 2x^3 + 10x^2

a. 2x^4 + 2x^3 + 10x^2 = 2x^2(x^2 + x + 5)

**complex algebra practice (substitutions): Solve for k: 3/(1 - (8/(7+K))) = 15

a. 3/A = 15 b. 3 = 15A c. A = 1/5 d. 1/5 = 1 - B e. B = 1 - 1/5 f. B = 4/5 g. 4/5 = 8/(7+K) h. 1/5 = 2/(7+K) I 7+K = 10 J K = 3

Inequality practice: 5x+7< 2x-2

a. 3x+7 < -2 b. 3x < -9 c. x < -3

**Absolute value inequalities practice: Express |x-7| ≤ 3 as an ordinary inequality.

a. 4 ≤ x ≤ 10

Factoring GCF from a binomial examples a. 5x+45 b. 9x^3 + 12x c. (x^7)(y^4) + (x^5)(y^6)

a. 5x+45 = 5(x+9) b. 9x^3 + 12x= 3x(3x^2 + 4) c. (x^7)(y^4) + (x^5)(y^6)= (x^5)(y^4)(x^2 + y^2)

Solve: 5x^2 - 10x-27= 13

a. 5x^2 - 10x- 40 = 0 b. 5(x^2-2x-8)= 0 c. 5(x-4)(x+2)= 0 d. divide by five: (x-4)(x+2)= 0 e. x=4 OR x=-2 (answer)

Simplify: dkfj a. 7x(x+2x) b. 5x(x^2+6x+12)

a. 7x^2+ 14x^2 b. 5x^3+30x^2+60x

Factor the following: a. x^2 + 14x + 24 b. x^2 + 4x - 21 c. x^2 - 2x - 35 d. x^2 -16x + 48

a. = (x+2)(x+12) b. = (x + 7)(x - 3) c. = (x -7)(x+5) d. = (x-12)(x-4); postive product & neg. sum

Solve the following (function notation) f(x) =x^2 + 4(x) -21 a. F(0) b. f(3) c. f(-1)

a. = -21 b. = 0 c. = -24

Simplify the following: 0.999951/0.993

a. = 1-0.000049/1-0.007 b. =1^2- 0.007^2/ 1-0.007 c. (1+0.007)(1-0.007)/1-0.007 d. 1.007 (answer)

Factor: dkfj a. 2x^2 - 22x + 48 b. 7x^4 - 56x^3 - 63x^2 (high intermediate) c. -3x^9 + 48xy^4 (advanced: nothing much harder than this)

a. =2(x-8)(x-3) b. = 7x^2(x-9)(x+1) c. = 3x(4y^2 + X^4)(2y+x^2)(2y-x^2)

Solve for x: dkfj a. 7x/6 + 2/3 = 13/2

a. Find LCM for fractions on both sides: 6 b. 7x+4=39 c. 7x=35 d. x=5 (answer).

Factoring Quadratics (example): x^2 + 8x +15

a. Find two numbers whose sum = 8 and whose product equals 15: 3 and 5 b. Factor therefore is the following: (x+3)(x+5)

System of equations (no solutions)

a. If the result has no variable and two numbers that do not equal each other, then there are no solutions. (parallel lines) b. If there are fewer equations than variables, you will not be able to solve fore the individual values of variables. c. Pay attention to wording. You may not need to solve everything.

Solve: kdfj x^2+5= 0

a. Impossible Note: you can't square something and get a negative number.

Problem: If y = 5+x and y = 12 - x, and if y^2 = x^2 + K, then K equals which of the following? a. 17 b. 25 c. 60 d. 119

a. K = y^2 - x^2 b. K = (y - x)(y + x) c. y-x = 5 d. y + x = 12 e. K = (5)(12) f: K = 60, letter C (answer)

Multiplying variables with powers

a. Simply add the powers: (x^a)(x^b)= X^(a+b)

Simplify the following: (y^2+2x-8)/(x-4)

a. [((y^2)/((x-4))+((2x-8)/x-4)} b....+ ((2(x-4))/(x-4) c.[((y^2)/((x-4)) + 2}

algebraic terms

a. constant: a number or symbol such as pi symbol that doesn't change in value. b. term: a product of constants and variables, including powers of variables. terms: 5, x, 6y^2, (x^5)(Y^6) c. coefficient: the constant factor of a term (6 is the coefficient of 6y^2). When no coefficient is written, the coefficient is 1. (See intro to algebra for more terms).

Solve for x and y using the elimination method: 2x+3y=15 x+2y= 11

a. multiply equation 2 by -2 and add both together to find y b. -2x-4y= -22 + 2x+3y=15 c. -y=-7 d. y=7 e. pluggin 7 back into one of the equation, we get x

Simplify: [(x/2)+(5/4)]/[(x/3) + (3/2)]

a. multiply numerator and denominator by LCM of all the smaller fractions denominators:12. b. (6x+15)/(4x+18) (answer)

Simplifying a complex fraction

a. multiply the numerator and denominator of the big fraction by the LCM of all the denominators of the little fraction

complex algebra practice (substitutions): Solve for x: (x^2 + 1)^2 - 15(x^2 + 1) + 50 = 0

a. n^2 - 15n + 50 = 0 b. (n - 10)(n - 5) = 0 c. n = 10 or n = 5 d. 10 = x^2 + 1 or 5 = x^2 +1 e. 9 = x^2 or 4 = x^2 f. x = +/- 3 or x = +/- 2

Solving algebraic equations

a. to solve for x follow the order of operations (GEMDAS) backwards.

Absolute values solve: |1+2x| = 4-x

a. x= 1 or x=-5 b. Plugging both back into the original equation, both numbers work so both answer are viable. Note: Both solutions will not always work when plugging them back in.

Solve the for x: dkfj a. 3x/5= 2/7

a. x= 10/21

Solve for x and y using the substitution method: 1. x+2y= 11 2. 2x+3y= 15

a. x= 11-2y b. 2(11-2y) + 3y= 15 c.22-4y+3y=15 d. 22- y= 15 e. 22= 15+y f. y = 7 g. x +2(7)= 11 h. x= -3 and y= 7

Absolute values solve |2x+5| = x+1

a. x=-4 or x=-2 b. Plug answers back into the right side of the original equation first c. -4+1= -3 (no because no absolute values can be negative. d. -2+1= -1 (no because no absolute values can be negative) e. there are no solutions.

Given the function f(x)= x^2+4x-21 Find the value(s) of x that would satisfy f(x) =24

a. x^2+4x-21= 24 b. x^2 + 4x - 45 = 0 c. (x+9)(x-5) = 0 d. x= -9 or x= 5 e. f(5) or f(-9)= 24 (answer)

Solve the following: x^2-3x-43=11

a. x^2-3x-54=0 b. (x-9)(x+6)=0 c. x-9= 0 d. x= 9 e. x+6= 0 f. x= -6 g. x= 9 or x= -6 (answer)

More examples of an even power of x is the square of another power a. x^6 b. x^8 c. x^6 - 16 d. x^7 - 4x^5 e. x^4 - 81 f. x^9 - x

a. x^6 = (X^3)^2 b. x^8 = (X^4)^2 c. x^6 - 16 = (x^3 - 4)(x^3 +4) d. x^7 - 4x^5 = x^5(x^2 -4)= X^5(x - 2)(x + 2) e. = (x^2 - 9)(x^2 + 9) = (x-3)(x+3)(x^2 + 9) f. = x(x^4+1)(x-1)(x+1)(x^2+1) Note for e and f: There is no way to factor a sum of squares, just a difference of squares.

Solve (three equations three unknowns): W-2x+3y= 13 2w+x-4y=-14 3w-x+2y=8

a. y=3 b. w=0 c. x= -2

Absolute values solve: |3x+2| + 1= 5

a. |3x+2|= 4 b. 3x+2= 4 or 3x+2= -4 c. x=2/3 or x=-2 (answer)

**Express the region -3 ≤ x ≤ 11 as an absolute value inequality

a. |x - 4| ≤ 7 answer find the average of the end points to get the answer.

Simplify the following:(2x^4-8x^2)/(x^2-5x-14)

a.[(2x^2(x^2-4))/... b. [(2x^2(x+2)(x-2))/(x+2)(x-7) c.[(2x^2(x-2))/(x-7)

Difference of two squares factor

a^2 - b^2 = (a+b)(a-c). example: x^2 - 49 = (x+7)(x-7)

If 5x+2y=55 and 2x-y= 19, then find the value of x+y.

answer is twelve. There is a shortcut if you recognize the there is a difference of three for both variables. multiply the second equation by negative one so that when you add them together, you get two coefficients of 3. Dividing by 3 on both sides eliminates the coefficient and give you the answer you need.

Simplify: (2x-5)(3x-1)

answer: (6x^2)-17x+5

Simplify: 15xy/3

answer: 5xy

Absolute Values: |x|= 5

answer: x= 5 or -5

Solve (no solutions): x-2y= 5 3x-6y

using substitution or elimination, you get two numbers that do not equal each other, so there are no solutions. (Parallel lines on a graph)