GRE Math 2

Age Question Word Problems

Usually easier to pick variable to represent the age right now, then use addition or subtraction to create expressions for ages at other times ex: Right now Steve's age is 1/2 Tom's age. In 8 years, twice Tom's age will be 5 more than 3 times Steve's age. How old is Tom now? S=1/2T -> T=2S 2(T+8) = 3(S +8) + 5 2((2S) + 8) = 3S + 29 4S + 16 = 3S + 29 S = 13 T = 2(13) = 26

Difference of Two Squares

(a + b)(a - b) = a^2 - b^2 ex: 9x^2 - 16 = (3x - 4)(3x + 4) ex: 25x^2 - 64y^2 = (5x + 8y)(5x - 8y) ex: x^2y^2-1 = (xy + 1)(xy - 1) ex: x^7 - 4x^5 = x^5(x^2 - 4) = x^5(x - 2)(x + 2) Practice: If y=5 + x and y=12 - x and if y^2=x^2 + k, what does k equal? y = 5+x -> y-x=5 y=12-x -> y+x=12 y^2=x^2 + k k = y^2 - x^2 = (y+x)(y-x) = (5)(12) = 60

Powers of 5 up to 5^4

5^1 = 5 5^2 = 25 5^3 = 125 5^4 = 625

Other Roots

x^2 = positive has 2 solutions x^2 = negative has no solution (positive)^3 = positive; x^3 has one positive solution (negative)^3 = negative; x^3 has one negative solution -With square roots, can only find square roots of positives, NOT negatives -Can take the cube root of any number on the number line, positive negative or zero ex: 3√8 = 2 3√0 = 0 3√-8 = -2 -Same positive and negative rules extend to even and odd roots -> n√ a n > or = 2 then for all n, n√0 = 0 and n√1 = 1 -> All roots preserve the order of inequality: If 0 < a < b < c then, 0 < n√a < n√b < n√c

Venn Diagram Word Problems

-2 groups and each member can belong to either group, both or none A + B + C + D = total Practice: Of the 100 students in a school, 60 are in the band and 35 are on the baseball team. If 25 students are neither in the band or on the baseball team. How many are in both? A + B= 60 B + C = 35 D = 25 A + B +C = 75 A + 35 = 75 -> A = 40 40 + B = 60 -> B = 20

√2

1.4

√3

1.7

1/p + 1/q = 1/f Solve for q

1/q = 1/f - 1/p 1/q = p/fp - f/fp 1/q = (p - f)/fp q = fp / (p - f)

√5

2.2

Word problems

If there are 3+ quantities in a word problem, you will always want to relate all of them to a single variable and construct a single equation

Sums of Sequences

The sum of a long sequence Sum = N(N + 1) / 2 Sum of a list = N * (a1 + an) / 2 a1= lowest term an = final term Ex: What is the sum of all multiples of 20 from 160 to 840 inclusive? 160 = 8*20 840 = 42*8 Inclusive counting: 42 - 8 + 1 = 35 Number of pairs = 17.5 Sum of list = 17.5 * (160 + 840) = 17.5 * 1000 = 17,500

Coefficient

Constant factor of a term; ex: 6 is the constant in 6y^2 ex: -1 is the constant in -x

Motion Question Word Problems

Distance = rate(speed) times time D = RT => R=D/T => T=D/R *If units are inconsistent then we need to change the units using a unit conversion Practice: A car moving at 72 km/hr moves how many meters in one second? (1 km = 1000 m) 1 hr = (60 min/hr)(60 seconds/min) = 3600 seconds R= D/T -> 72 km/hr = 72,000 m/3600 s = 720/36 =20 m/s

Multiplying Expressions

Distributive Law allows for multiplication to distribute over addition and subtraction A(B+C) = AB + AC *Does NOT distribute over multiplication 3*(xy) = 3xy -If we multiply a number (a constant) times a monomial with a variable, the constant multiples the coefficient ex: 2*(r^4*s^2*t^3) = 2r^4*s^2*t^3 *Multiplying powers means adding the exponents

How to eliminate fractions

Eliminate fractions by multiplying by the LCM ex: x/2 + 5/4 = x/3 + 3/2 LCM = 12 12(x/2) + 12(5/4) = 12(x/3) + 12(3/2) = 6x + 15 = 4x +18 -> 2x = 3 -> x= 3/2 *Can simplify a complex fraction by multiplying both the numerator and denominator of a fraction by the LCM of all the denominators of the little fractions

Exponential Equations

If two powers with the same base are equal then the exponents must be equal b^x=b^y => x=y ex: 49^x = 7^6-x => (7^2)^x = 7^6-x => 7^2x = 7^6-x => 2x = 6-x => 3x = 6 => x=2 ex: (5√3) ^3x + 7 = 3^2x => (3^1/5)^3x+7 = 3^2x 3^(3/5x + 7/5) = 3^2x => 3/5x + 7/5 = 2x => Multiply each side by 5 => 3x + 7 = 10x => x=1 Can rewrite the bases so they are a power of the other ex: 27^(2x-2) = 81^(x+1) => (3^3)^(2x-2) = (3^4)^(x+1) => 3(2x-2) = 4(x+1) => 6x-6 = 4x+4 => x=5

Laws of Exponents II

Just as multiplication distributes over addition and subtraction (P*(M+N) = PM + PN), exponents distribute over multiplication and division (ab)^n = (a^n)(b^n) (a/b)^n = a^n/b^n ex: 18^8 = (2*3^2)^8 = 2^8*3^16 -ILLEGAL to distribute exponent over addition or subtraction; must solve in parenthesis first (M + N)^p =/= M^p + N^p -Lower power is always a factor of a higher power ex: 3^32 - 3^28 = (3^28)(3^4) - 3(28)(1) = (3^28)(3^4 - 1) = (3^28)(81-1) = 80(3^28) - b^s = b^t => s=t

Multiple Traveler Word Problems

-each traveler of each trip gets its own D=RT equation -More often you will have to use the techniques for solving 2 equations with 2+ unknowns (elimination and substitution) ex: Frank and George started traveling A to B at the same time. G's constant speed was 1.5 times Frank's constant speed. When G arrived at B, he turned back immediately and returned by the same route. He crossed paths with Frank who was coming toward B, when they were 60 miles away from B. How far away are A and B? Frank: D - 60 = RT D + 60 = 1.5R*T D + 60 = 1.5(D - 60) => D + 60 = 1.5D - 90 => 150 = .5D => 300 = D

Powers and Roots

0^n = 0 -> a negative to any even power is positive and a negative to any odd power is negative -> x^2 = 4 has 2 solutions, x = 2 or x = -2 because either equals +4 -> By contrast x^3 = 8 only has one solution, x = 2 -> Something squared even equals a negative has no solution (x - 1)^2 = -4 no solution -> Something cubed odd equals a negative does have a solution (x - 4)^3 = -1 => x - 4 = -1 => x = 3

3 Equations with 3 unknowns

1) Pick 2 of 3 equations and using substitution or elimination, eliminate 1 variable 2) pick another pair of original equations and eliminate the same variable 3) Now use 2 equation with 2 unknown technique 4) Plug into any original equation to find the value of the third variable ex: A) w - 2x + 3y = 13 B) 2w + x - 4y = -14 C) 3w -x +2y = 8 B) 2w + x - 4y = -14 + C) 3w - x + 2y = 8 => D) 5w - 2y = -6 => 5w =2y - 6 A) w - 2x + 3y = 13 + B) *2 => 4w + 2x -8y = -28 => 5w -5y = -15 => 5w = 5y - 15 => 2y - 6 = 5y - 15 => 3y = 9 => y=3 => 5w = 2(3) - 6 => 5w = 0 => w=0 => 0 - 2x + 3(3) = 13 => -2x = 4 => x= -2

Powers of 2 up to 2^9

2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 16 2^5 = 32 2^6 = 64 2^7 = 128 2^8 = 256 2^9 = 512

Powers of 3 up to 3^4

3^1 = 3 3^2 = 9 3^3 = 27 3^4 = 81

Powers of 4 up to 4^4

4^1 = 4 4^2 = 16 4^3 = 64 4^5 = 256

Constant

A number or symbol such as pi which doesn't change in value

Term

A product of constants and variables, including powers of variables ex: 5, x, 6y^2, x^5y^6z^7

Simplifying expressions

Add like terms. Multiplication is commutative so orders of factors in multiplication doesn't matter ex: 5xy + 7yx = 12xy *Can't add terms of different variables or powers *When a subtraction sign appears you must change every sign inside the parenthesis to its opposite ex: (x^3 - 3x^2) - (x^3 + 3x^2) = x^3 - 3x^2 - x^3 - 3x^2

Elimination Method

Always allowed to add 2 equations: 7x + 3y = 5 + 2x - 3y = 13 => 9x = 18 => x=2 Plug into either equation to solve for y 2x - 3y = 13 -> 2(2) - 3y = 13 -> -3y = 9 -> y=-3 *Strategy is multiply both sides of 1 equation by 1 number and both sides of another equation by another number so that for 1 of the variables, the coefficients are equal and opposite ex: 2x + 3y = 15 & x + 2y = 11 => -2(x + 2y) = -2(11) 2x + 3y = 15 + -2x - 4y = -22 = -y = -7 -> y=7

Sequence Patterns

An = n is the sequence of all positive integers An = 2n - 1 is the sequence of all positive odd numbers An = 7n is the sequence of all positive multiples of 7 An = n^2 is the sequence of all perfect squares An = 3^n is the sequence of all powers of 3

Factoring Quadratics

Factor : x^2 + 8x + 15 Need to find two numbers whose sum=8 and whose product=15 =(x+3)(x+5)

Infinitely many solutions

One equation is a multiple of the other -> will wind up with an always true equation such as 7=7

Stranger Operators

Remember to find the numerical value of the expression in side parenthesis first

Extraneous solutions

Solutions that result correctly from the math but which won't work in the original equation ex: |2x + 5| = x + 1 2x + 5 = x+ 1 or 2x + 5 = -x - 1 x = -4 x = -2 |2(-4) + 5| = -4 + 1 |2(-2) + 5| = (-2) + 1 |-3| = -3 |1| = -1 3 =/= -3 1 =/= -1 Neither solution works

No solutions

Started with the system of equations and led us to a NEVER TRUE equation ex: x - 2y = 5 => x=2y + 5 & 3x - 6y = 8 3(2y + 5) - 6y = 8 => 6y + 15 - 6y = 8 => 15 = 8 *Same expression equals 2 different things means that there are parallel lines x - 2y = 5 -> (Mult. by 3) 3x - 6y =15 & 3x - 6y=8

How to solve Quadratic Equations

ax^2 + bx + c = 0 *can have 2 solutions, 1 solution or no solution 1) get everything on one side of equation and set equation = to zero 2) Divide by any GCF 3) Factor 3) Use zero product property to create 2 linear equations and solve ex: 3x^2 - 9x -70 = 14 -> 3x^2 - 9x - 84 = 0 divide by 3 x^2 - 3x - 28 = 0 -> (x-7)(x+4) = 0 -> x=7 or x=-4

Negative Exponents

b^-n = 1/b^n ex: 13^4 / 13^7 = 13^-3 = 1/13^3 -A negative exponent on a fraction will be the reciprocal to the positive power (p/q)^-n = (q/p)^n -A negative power in the numerator of a fraction can be moved to the denominator as positive power and vice verse ex: b^5*d^-8 / h^-4*k^7 = b^5*h^4 / d^8*k^7 Practice: (1/3) ^ -8 = 3^8 3^-3 = (1/3)^3

Absolute Value Equations

ex: |x| = 5 has 2 solutions, could be +5 or -5 ex: |2x + 3| = 5 2x + 3 = 5 -> x= 1 or 2x+3 = -5 -> x = -4 *If the absolute value is not by itself on one side then we will have to isolate it ex: |3x + 2| +1 = 5 -> |3x + 2| = 4 3x+2 = 4 -> x = 2/3 or 3x+2 = -4 -> -2

Square of a Difference

(a-b)^2 = (a-b)(a-b) = a^2 - ab -ba + b^2 = a^2 -2ab + b^2

Square of a Sum

*Common mistake = illegal to distribute an exponent across addition or subtraction illegal: (a + b)^2 =/= a^2 + b^2 legal: (a + b)^2 = (a+b)(a+b) = a^2 + ab + ba +b^2 = a^2+2ab+b^2 This is called the Square of a Sum

Average Speed Word Problems

*NOT solved by adding numbers and dividing by 2 Average velocity = total distance / total time *Often need to find the total distance/total by adding the distances and times of the individuals legs *Sometimes the problems won't give you all the #'s Practice: Cassie drove from A to B at constant speed 60 mph. She then returned on the same route, from B to A at constant speed 20 mph. What was her average speed? T1 = D/R = D/60 T2 = D/R = D/20 Tt = T1 + T2 = D/60 + D/20 = multiply by the LCM = D/60 + 3D/60 = 4D/60 = D/15 Vaverage = Dt/Tt = 2D/(D/15) = 2D/1 *15/D = 30 mph

Inequalities

*When you see "number" or see x, you have to think of all numbers - positive, negative, zero, integers, fractions and decimals *Can always add or subtract the same thing from both sides and the inequality remains the same ex: x + 7 > 2 => x > -5 *Multiply or divide both sides by any positive number will preserve the inequality *Multiply or divide both sides by a negative number will reverse the inequality ex: 2/x > 1/3 x=/= 0, x>0 Because all #'s are positive we can cross multiply 6>x ex: -4 < 5 -3x < 17 Subtract 5 from all 3 parts -9 < -3x < 12 Divide -3 from all 3 sides -3 > x > -4

Rationalizing

-Avoid radicals in the denominator ex: 14/√21 * √21/√21 = (14√21)/21 = (2√21)/3 ex: 4 - √6 / 2√3 *√3/√3 = 4√3 - (√6)(√3) / 2*3 = 4√3 - 3√2 / 6 Multiply by the opposite ex: (4 + 2√5) / (3 + √5) * (3 - √5)/(3 - √5) = (4 + 2√5)(3 - √5) / 9 - 5 = (12 - 4√5 + 6√5 - 10) / 4 = (2 - 4√5 + 6√5) / 4 = (2 - 2√5) / 4 = 1 + √5 = 4

Operations with Roots

-Can combine 2 terms ONLY if they have the same radical factor ex: √72 - √32 = √(36*2) - √16*2) = 6√2 - 4√2 = 2√2 ex: 6√2 - 4√3 Can't be simplified further because the radicals aren't equal -Multiplication is commutative and and associative (can switch the order around). Can multiply the whole by whole and radical by radical ex: (3√5)(2√15) = 6√(5*15) = 6√(5*5*3) = 6*5√3 = 30√3 -Also when we divide, we can divide whole by whole and radicals by radicals ex: 54√35 / 18√5 = 54/18 * √35/√5 = Simplify fractions 3√7 -If we are squaring a radical expression, we square the number and radical separately ex: (2√3)^4 = 2^4 * (√3)^4 = 16 * √3 * √3 * √3 * √3 = 16 * 3 * 3 = 144 OR ((2√3)^2)^2 = (4*3)^2 = 12^2 = 144 -Any even power of a square root can be written as a power of a whole number ex: (√2)^48 = ((√2)^2)^24 = 2^24

Comparing inequalities

-Can combine inequalities (a<b, b<c) in the same direction (a<b<c) -Can add inequalities in the same direction a<b & c<d then (a+c) < (b+d) ex: 5>2 and 11>8 then 5+11 = 16 > 2+8=10 -Can subtract inequalities in opposite direction a>b and d<c then (a-d)>(b-c) ex: 20>15 and 10<12 then 20-10 = 10 > 15-12 = 3 -No general rule for the multiplication or division of inequalities -any positive > any negative -Adding a positive makes a number greater, subtracting a positive makes it less

Recursive Sequences

-Can't jump right away to value, have to work out way term by term to the value we want. Some sequences depend on the previous two terms Practice: For a sequence with rate An = An-1 + An-2 for n>2, and starting values of A1 = 1 and An = 3, find value of A6. A1 = 1 and A2 = 3 A3 = A2 + A1 = 3 + 1 = 4 A4 = A3 + A2 = 4 + 3 =7 A5 = A4 + A3 = 7 + 4 = 11 A6 = A5 + A4 = 11 + 7 = 18

The Units Digit Question

-Focus on the single digit multiplication only -Look for the repeating pattern and determine the period of the pattern (period if often 4) -Extend the pattern using multiples of the period ex: What is the units digit of 57^123? 7^1 = 7 7^2=_9 (9*7=63) 7^3 = __3 (3*7 = 21) 7^4 = __1 (7*1 = 7) 7^5 = __7 Answer: 1

Exponential Growth

-For a positive base greater than 1, the powers get larger at a fast rate ex: 7^5 = 16,801 -For a positive base less than 1, the powers get smaller at a fast rate ex: (1/2)^8 = 1/256 -For a negative base less than -1, the absolute values are getting bigger each time, but the + or - signs are alternating ex: (-3)^5 = -243 (-3)^6 = +729 -For a negative base between -1 and 0, the absolute values are getting smaller but the + or - signs are alternating ex: (-1/2)^7 = -1/128 (-1/2)^8 = +1/256

Advanced Numerical Factoring

-Helps find the prime factorization of large numbers ex: 2491 = 2500 - 9 = (50^2) - (3^2) = (50 + 3)(50 - 3) = 47x53 ex: 9975 = 10,000 - 25 = (100^2) - (5^2) = (100 + 5)(100-5) = 105x95 = (5x21)(5x19) = 5x7x3x19 *Can also help with decimals Ex: simplify (0.999856/0.998) - 1 = (1-0.000144/1-0.012)-1 = (1-0.012)(1+0.012)/(1-0.012)-1 =(1 + 0.012) - 1 = 0.012

Comparing the size of different roots

-If b>1 and if n>m then 1 < n√b < m√b < b The higher the root, the smaller the number ex: 19 > √19 > 3√19 > 1 ex: 20√19 > 30√19 -If 0 < b < 1, and if n>m, then 0 < b < m√b < n√b < 1 ex: 2/5 < 50√2/5 < 75√2/5 < 1 The HIGHER the root, the CLOSER to 1

Square Roots continued

-If squaring makes numbers bigger, then taking a square root must make them smaller 1 < b^2, then b < b^2 => √b < b -If squaring makes numbers smaller, then taking a square root must make them bigger 0 < b < 1, then b > b^2 => √b > b

Square Roots

-If test makers write √ consider the positive roots only -If your calculations lead to a variable squared, consider both positive and negative roots √0 = 0 -CAN'T take the square root of a negative -For A > or = to 0, √A > or = to 0 -If the test writes "solve the equation" x^2 = 5, there are 2 solutions, both positive and negative √5 -If A<B<C then √A<√B<√C -Square roots are always positive regardless of whether y is positive or negative √y^2 = |y| -> x^2 = k => x = + or - √k Practice: If (x - 3)^2 = 16 Solve for x √(x - 3)^2 = √16 x - 3 = +4 => x = 7 OR x - 3 = -4 => x = -1

Shrinking and Expanding Gaps Word Problems

-Involve 2 travelers moving to or away from each other -Two travelers moving in opposite directions: *When moving in opposite directions, always ADD the speeds -If the 2 travelers are approaching each other, the sum of the speeds is the speed at which the gap is shrinking -If the 2 travelers are moving away from each other, the sum of the speeds is the speed at which is the gap is expanding -Two travelers moving in the same direction: *When moving in the same direction, we always SUBTRACT the speed -If the faster traveler is in front, then the difference in which the gap is expanding -If the slower traveler is in front, then the difference in speeds is the speed at which the gap is shrinking *In a problem where the gap is obviously shrinking or expanding, sometimes saves time to set up a D=RT for the gap itself Practice: Car X & Y are traveling from A to B on the same route at constant speeds. Car X is initially behind Car Y but Car X's speed is 1.25 times Car Y's speed. Car X passes Car Y at 1:30 pm. At 3:15 pm Car X reaches B and at that moment Car Y is still 35 miles away from B. What is Car X's speed? Time interval: 1:30 to 3:15pm => 1/2 + 1 + 1/4 = 2/4 + 5/4 = 7/4 hr Gap: R = D/T = 35 miles/(7/4 hr) = 35 * 4/7 = 20 mph x = 1.25y x - y = 20mph 1.25y - y = 20 = > 1/4y = 20 => y=80 mph x = 1.25(80) = 100 mph

Work Word Problems

-Machines or workers and how fast they can get certain jobs done A = RT a= amount of work done R= the work rate t=time 2 categories of work problems 1) Using proportions Practice: A machine, working at a constant rate, manufactures 36 staplers in 28 minutes. How many staplers can it make in 1 hour 45 minutes? 1 hour 45 minutes = 60 + 45 = 105 minutes staplers/ time: 36/28 minutes = x/105 minutes *Don't cross multiply 9/7 = x/105 => 9 = x/15 => 135 staplers = x 2) Multiple machines/workers working at different rates and how much they can get done together *the SUM of the individual work rates *Need to match units on each side Practice: When Amelia and Brad detail a car together, 1 car takes 3 hours. When Amelia details a car alone, 1 car takes 4 hours. How long does it take Brad to detail a car alone? Rate = car/hours Rab = Ra + Rb Rab = 1/3 Ra = 1/4 Rb = Rab - Ra = 1/3 - 1/4 = 4/12 - 3/12 = 1/12 It takes Brad 12 hours to detail 1 car

Mixture Question Word Problems

-Mixing solutions of various concentrations. Many materials dissolve in water which forms a solution. -The dissolved substance is called the solute -Concentration indicates how strong the solution is; how much solute is dissolved in a given quantity of water; always expressed as a percent Concentration = Amt of solute/ amt of solution * 100 -Add water to a solution -> makes a less concentrated solution -Add pure solute to a solution -> makes a more concentrated solution -If the amounts of 2 different solutions are initially unknown, we have to set up simultaneous equations. One equation will be a "total" equation and the other will be the amount of solute. Practice: State with unlimited supplies a 20% H2S04 solution and of a 50% H2S04 solution. We combine x liters of the first with Y liters of the second to produce 7 liters of a 40% H2S02 solution. What does x equal? total equation = x + y = 7 -> y = 7 - x Solute = .4 * 7 = 2.8L Solute #1 = 0.2x Solute #2 = 0.5Y => 0.2X + 0.5Y = 2.8L => 0.2X + 0.5(7 - x) = 2.8 => 2x + 5(7 - x) = 28 => 2x + 35 - 5x = 28 => -3x = -7 => x = 7/3 L

Laws of Exponents

-Multiplying 2 powers of the same base means you can add the exponents (7^m)*(7^n) = 7^m+n -Dividing powers of the same base means you can subtract the exponents a^m / a^n = a^m-n -a^0 = 1 if a=/= 0 -Raising a power to a power results in multiplying the exponents (a^m)^n = a^m*n -CAN'T apply these rules if bases are different ex: 2^3*3^5 -NO LAW for the sum or difference of powers ex: 3^4 + 3^7 or 5^8 - 5^2

Fractional Exponents

-Raising a number to the 1/2 is the same as squaring it b^1/2 = √b ex: 2^1/2 = k => (2^1/2)^2 = k^2 => 2 = k^2 => √2 = k The rules: b^1/m = m√b b^m/n = (b^m)^1/n = (b^1/n)^m These are true for all positive numbers. If the denominator of the exponent-fraction is odd then the base can be negative as well ex: 2^3/5 = (2^3)^1/5 = 5√(2^3) = 5√8 OR 2^3/5 = (2^1/5)^3 = (5√2)^3 ex: 8^4/3 = 3√8^4 = (3√8)^4 = 2^4 = 16

Simplifying Roots

-Simplify square roots by factoring out the largest perfect square root ex: √75 = √(25*3) = 5√3 -Find the prime factorization: any pairs of prime factors and any even powers of primes are perfect squares ex: √2800 = √(28*100) = 10√(4*7) = 20√7 ex: √(3^5) = √(3^2 * 3^2 * 3^1) = 3^2√3

The Properties of Roots

-Some properties of exponents are the same as the properties of roots: -Roots also distribute over multiplication and division √PQ = (√P)(√Q) √(P/Q) = √P/√Q ex: Simplify (√12)(√27) = √12*27 = √12*3*9 = √36*9 = √36 * √9 = 6*3 = 18 ex: √(4/50) = Reduce the fraction √(2/25) = (√2)/5 -Roots do NOT distribute over addition and subtraction √P + √Q =/= √(P+Q)

Equations with Square Roots

-Undoing a square root by squaring both sides - unsquaring could be positive or negative -Radical equations can have extraneous roots, must plug back in to make sure they work ex: √(x + 3) = x - 3 => x + 3 = (x - 3)^2 => x+3 = x^2 - 6x + 9 = > 0 = x^2 -7x + 6 => 0 = (x - 6)(x -1) x = {1, 6} Plugged back in √(1 + 3) = 1 - 3 => 2 =/= -2 Doesn't work √(6 + 3) = 6 - 3 => 3 = 3 Works ex: √(2x - 2) = √(x - 4) => 2x - 2 = x - 4 => x = -2 Plugged back in: √(2(-2) - 2) = √(-2 - 4) => √-6 = √-6 Can't take the square root of a negative so no solution -MUST isolate the radical one side so it would make sense to square both sides: Ex: 2+ √(4 - 3x) = x => √(4 - 3x) = x - 2 => 4 - 3x = (x - 2)^2 => 4 - 3x = x^2 - 4x + 4 => 0 = x^2 - x => 0 = x(x - 1) => x = 0, x = 1

Inclusive Sequences

-Used when both the starting value and ending value are included in what we are counting ex: days used for workshop -Perform ordinary subtraction and add one ex: How many multiples of 8 are there from 200 to 640 inclusive? 8*25 = 200 8*80 = 640 200 is the 25th multiple of 8 and 640 is the 80th multiple of 8 and both are included number = 80 - 25 + 1 = 56

Cubes of 1 - 10

6^3 = 216 7^3 = 343 8^3 = 512 9^3 = 729

Expression

A collection of one or more terms joined by addition or subtraction. *Don't have equal signs ex: y^2 - x^2

Arithmetic Sequence

An evenly spaced list Common difference: the fixed amount we add to each term to get the next An = A1 + (n - 1)*d Practice: Let x be the set of all positive integers that when divided by 8, have a remainder of 5. What is the 76th number in the set? An = 5 + 8*(n - 1) A16 = 5 + 8(75) = 605

Substitution Method

Solve one equation, either one, for one of the variables. Get one variable by itself on one side of the equation: x + 2y = 11 - > x = 11 - 2y Now replace x in the other equation with the expression that x equals: 2x + 3y = 15 -> 2(11-2y) + 3y = 15 - > 22 - 4y + 3y = 15 -> 22 - y =15 -> y= 7 Now plug the value of x back into equation to solve for x: x= 11 - 2(7) = 11 - 14 = -3

Proportional Reasoning Steps

a) Pick easy numbers to satisfy the equation, can also change any constants to 1 b) Change whatever values need to be changed, leave the quantity in the Q as unknown and solve for it Ex: In V^2 = 2ad, if V triples and a doubles, then d is multiplied by what? 1^2 = (1)(1)d 3^2 = (1)(2)d => 9 = 2d => 9/2 = d ex: In T^2 = KR^3, if T is multiplied by 5, R is multiplied by what? (1)^2 = (1)R^3 => 5^2 = R^3 => 3√25 = R R = 3√25 = 3√(5^2) = 5^2/3

Function notation

f(x) "x" is the input to the function "f" ex: Given the function f(x) = x^2 + 4x - 21 Find the values of x that would satisfy f(x) = 24 f(x) = x^2 + 4x - 21 = 24 x^2 + 4x - 45 = 0 -> (x+9)(x-5) = 0 -> x=-9, x= 5 ex: Given the function f(x) = x^2 +kx + 4 Find the value of k if f(2) = 18 (2)^2 + k(2) + 4 = 18 -> 8 + 2k = 18 -> k=5

Absolute Value Inequalities

|x| is the distance of x from zero |x - 5| is the distance of x from +5 |x + 3| is the distance of x from -3 Ex: Express |x-7| < 3 as an ordinary inequality 7 - 3 = 4 7+3 =10 4<x<10 ex: Express the region -3<x<11 as an absolute value inequality *Middle of average endpoints (11+-3)/2 = 4 X can be as far as 7 above 4 or 7 below 4 |x - 4| < 7