Divisible by 2 if
Even
Divisible by 3 if
Sum of the integer's digits are divisble by 3
Divisible by 4 if
integer is divisible by 2 twice or the last two digits are divisible by 4
Divisible by 5 if
integer ends in 5 or 0
Divisible by 6 if
integer is divisible by both 2 and 3
Divisible by 8 if
Integer is divisible by 2 three times or if the last three digits are divisible by 8
Divisible by 9 if
sum of the integer's digits are divisble by 9
Divisible by 10 if
integer ends in 0
Fewer Factors More Multiples
Factors divide into an integer and are therefore less than or equal to that integer. Positive multiples on the other hand multiply out from an integer and are therefore greater than or equal to that integer
Add/Subtract two multiples
results is a multipleMultiple of 3 + Multiple of 3 = Multiple of 3
Add/Subtract a multiple with a nonmultiple
result is a nonmultipleMultiple of 3 + Nonmultiple of 3 = Nonmultiple of 3
Add/Subtract two nonmultiples
results can be multiple or nonmultiple
Property of GCF and LCM
(GCF of m and n) x (LCM of m and n) = m x n
Property of GCF and LCM
The GCF of m and n cannot be larger than the difference between m and n. For exmaple, assume the GCF of m and n is 12. Thus m and n are both multiples of 12. Consecutive multiples of 12 are 12 units apart on the number line and therefore cannot be less than 12 units apart
Property of GCF-consecutive multiples
have a GCF of n.For example, 8 and 12 are consecutive multiples of 4. Thus 4 is a common factor of both numbers. But 8 and 12 are exactly 4 units apart. Thus 4 is the greatest possible common factor of 8 and 12
Factors of perfect squares
All perfect squares have an odd number of total factors. Vice versa so if a integer has an odd number of total factors it must be a perfect square
Prime Factorization Factors
the prime factorization of a perfect square contains only even powers of primes. Vice versa.
Factorials and Divisibility
Because it is a product of all the integers from 1 to N, any factorial N! must be divisible by all integers from 1 to N. N! is a multiple of integers from all the integers from 1 to N
Range of possible remainders
When you divide an integer by 7, the remainder could be any number between 0 and 6 inclusive. Notice that you cannot have a negative remainder or remainders larger than 7. There are exactly 7 possible remainders.
Arithmetic with remainders
You can add and subtract remainders as long as you correct excess or negative remainders
Multiplication with remainders
You can multiply remainders as long as you correct excess remainders at the end. For example, if x has a remainder of 4 upon division by 7 and z has a remainder of 5 upon division by 7, then 4 x 5 gives 20. Two additional 7's can be taken out of this remainder, so xz will have remainder 6 upon division by 7.
Factor Counting
Solve factor counting problems by writing the prime factorization in exponential form, adding 1 to all of the exponents and multiplying
Add/Subtract Odds & Evens
ODD +_ EVEN = ODDODD + _ ODD = EVENEVEN + _ EVEN = EVEN
Multiply/Divide Odds & Evens
ODD x ODD = ODDEVEN x EVEN = EVEN (and divisible by 4)ODD x EVEN = EVEN
Representing Evens and Odds Algebraically
Even numbers can be written as 2nOdd numbers can be written as 2n+1 or 2n-1
Odds / evens with multiple variables
Approach by testing different odds/evens for each variable
Sum of two primes
Unless one of the numbers is 2, it will result in an even number
Absolute value of a difference
Ansolute value of x-y can be interpreted as the distance between x and y.For example, rephrase the absolute value of x-3<4 as "the distance between x and 3 is less than 4
Multiplying & Dividing Signed Number
the result will be positive if you have an EVEN number of negative numbers in the collection. The result will be negative if you have an ODD number of negative numbers.
Disguised + or - questions
Whenever you see >0 or <0, think Positives & Negatives
Complex Absolute Value
For a problem with two different variables, generally without constants, are more easily solved using a conceptual approach rather than algebraic. Try picking and testing numbers, specifically positives, negatives and zero
Complex Abs Value with more than one expression
an absolute value with more than one expression but only one variable and one or more constants is usually easier solved with an algebraic approach
Evenly Spaced Sets
All sets of consecutive integers are sets of consecutive multiples. All sets of consecutive multiples are evenly spaced sets. All evenly spaced sets are fully defined if the 3 parameters are known: (1) first and last numbers in the set, (2) the increment, (3) the number of items in the set
Properties of Evenly Spaced Sets
(1) Mean and the median are equal, (2) mean and median of the set are equal to the average of the FIRST and LAST terms, (3)the sum of the elements in the set equals the arithmetic mean x number of items in set
Counting Integers for consecutive integers (how many integers from x to y?)
Add one before you are doneFor example, how many integers between 6 and 10? Count 6, 7, 8, 9, 10 or just subtract 10-6 + 1(Last - First + 1)
Counting Integers for consecutive multiples (how many integers from x to y?)
(Last - First) / Increment + 1
Sum of Consecutive Integers
(1) Average the first and last term to find the middle of the set(2) Count the number of terms(3) Multiply the middle term by the number of terms to find the set
Facts about sums and averages of evenly spaced sets
(1) the average of an ODD number of consecutive integers will always be an integer(2) the average of an EVEN number of consecutive integers will never be an integer
Products of Consecutive Integers and Divisibility
the product of n consecutive integers is divisible by n!
Sums of Consecutive Integers and Divisibility
The sum of n consecutive integers is divisible by n if n is odd, but it is NOT divisible by n if n is even
Beware of even exponents
They hide the original sign of the base. Any base raised to an even power will result in a positive answer
Fractional Base Exponents
While most positive numbers increase when raised to a higher exponent, numbers between 0 and 1 decrease
Adding Exponents
When multiplying two terms with the same base, combine exponents by adding
Subtracting Exponents
When dividing two terms with the same base, combine exponents by subtracting
Nested Exponents
When raising a power to a power, combne exponents by multiplying
Negative Exponents
Raising a number to a negative exponent is the same as raising the number's reciprocal to the equivalent positive exponent
An Exponent of 1
When you see a base without an exponent, write in an exponent of 1
An exponent of 0
Any nonzero base raised to the power of zero is equal to 1
Fractional Exponents
Within the exponent fraction, the numerator tells us what power to raise the base to, and the denominator tells us which root to take
When to Simplify Eponential Expressions
(1) You can only simplify expressions that are linked by multiplication or division (not addition or subtraction)(2) You can simplify expressions linked by multiplication or division if they have either a base or an exponent in common
Sum of consecutive integers and divisibility
For any set of consecutive integers with an ODD number of items, the sum of all the integers is always a multiple of the number of items. For example: 4+5+6+7+8= 30 which is a multiple of 5For any set of consecutive integers with an EVEN number of items, the sum of all the integers is never a multiple of the number of items.
Exponents Strategy
1) Simplify or factor any additive or subtractive terms2) Break every non-prime base down into prime factors3) Distribute the exponents to every prime factor4) Combine the exponents for each prime factor and simplify
Signs of Square Roots
Even roots only have a positive value. A root can only have a negative value if (1) it is an odd root and (2) the base of the root is negative
Simplifying Square Roots
You may only seperate or combine the product or quotient of two roots. You cannot seperate or combine the sum or difference of two roots
Estimating Roots of Imperfect Squares
If there is no coefficient in front you may estimate by figuring the two closest perfect squares on either side of it. If you want to estimate a square root with a coefficient, simply estimate the square root and then multiply by the coefficient. Or combine the coefficient with the root.
Simplifying Roots with Prime Factorization
1) Factor the number under the radical sign into primes2) Pull out any pair of matching primes from under the randical sign, place one of them outside the root3) Consolidate the expression
Stategies to solve data sufficiency - rephrase
Rephrase: take the given information and reduce to its simplest form then focus on how the piece of info relates to the question
Strategy for solving data sufficiency- type of problem
Is it a value question or a yes/no question? Never turn a yes/no question into a value question
Strategy for solving data sufficiency-test numbers
For a value question, try to find multiple answers. For a yes/no question try to find a maybe.Be sure to try a positive, negative, integera and fractional number unless explicitly told otherwise.
Powers of 10- multiply by positive power of 10
Move decimal forward (right) to make the positive number larger. For example: 3.9742 x 10^3 = 3974.2 (move decimal forward 3 spaces)89.507 x 10 = 895.07 (move decimal forward 1 space)Add zeros if needed: 2.57 x 10^6 = 2,570,00014.29 / 10^5 = 0.0001429
Powers of 10- divide by positive power of 10
Move decimal backward (left) to make the positive number smaller. For example: 4169.2 / 10^2 = 41.692 (move backwards 2 spaces)89.507 / 10 = 8.9507 (move backwards 1 space)
Powers of 10-negative powers
Negative powers reverse the process. Ex: 6782.01 x 10^-3 = 6.78201 (Moving the decimal forward by negative 3 spaces means moving it backward by 3 spaces)53.0447 / 10^-2 = 5304.47 (Moving the decimal backward by negative 2 spaces means moving it forward by 2 spaces)
The Last digit Shortcut
When asked to find the units digit, just look at the units digit of the product. For example: What is the units digit of (7^2)(9^2)(3^3)? Step 1: 7 x 7 = 49 - Drop all except units digit - 9Step 2: 9 x 9 = 81 - Drop all except units digit - 1Step 3: 3 x 3 x 3 = 27 - Drop all except units digit - 7Step 4: 9 x 1 x 7 = 63The units digit of the final product is 3
Heavy Division Shortcut
Used for large numbers. Example: What is 1,530,794 / (31.49 x 10^4) to the nearest whole number?Step 1: Set up the division problem in fraction formStep 2: Rewrite the problem, eliminating powers of 10: 1,530,794 / 314,900Step 3: The goal is to get a single digit to the left of the decimal in the denominator. Just remember whatever you do to the denominator, to do to the numerator. 1,530,794 / 314,900 = 15.30794 / 3.14900Now you have the single digit 3 to the left of the decimal in the denominatorStep 4: Focus only on the whole number parts and solve.15.30794 / 3.14900 is approx 15/3 which = 5
Add/Subtract Decimals
Line up the decimal points!
Multiply Decimals
Ignore the decimal point until the end. Just multiply the numers as if they were whole numbers. Then count the total number of digits to the right of the decimal point in the factors. In the factors, count all the digits to the right of the decimal point--then put that many digits to the right of the decimal point in the productIf multiplying a very large number and a very small number, move the decimals in the opposite direction the same number of places.
Powers and Roots
(0.04)^3 = 0.000064 --> (0.04)^3 has 2 places so 2 places x power of 3 = 6 places in the final answer(0.000000008) ^(1/3) = 0.002 --> (0.000000008)^(1/3) has 9 places so 9 places /power of 3 = 3 places in the final answer
Teminating Decimals
Only have prime factors of 2's and 5's. If there are other prime factors, it is not a terminating decimal.
Repeating Decimals
divide any number by 9 and it becomes a repeating decimal. For example: 4/9 = 0.44444444 forever. 3/11 = 27/99 = 0.272727272727 forever
Numerator and denominator rules for positive numbers
As numerator goes up, the fraction increases.As denominator goes up, the fraction decreases.Adding the same number to both numerator and denominator brings the fraction closer to 1 regardless of the fraction's value. If the fraction is originally smaller than 1, the fraction increases in value as it approaches 1. if the fraction is originally larger than 1, the fraction decreases in value as it approaches 1.
No addition or subtraction shortcuts with fractions
1)find a common denominator2) change each fraction so that it is expressed using this common denominator3) add up the numerators only
Fraction Operations:Funky Results
For proper fractions:Adding Fractions --> Increases the valueSubtracting Fractions --> Decreases the valueMultiplying Fractions --> Decreases the valueDividing Fractions--> increases the value
Comparing Fractions
Cross MultiplyCross multiply the fractions and put each answer by the corresponding numeratorFor example: 7/9 vs. 4/5 (7 x 5) = 35(4 x 9) = 36Put 35 next to corresponding 7/9 and 36 next to corresponding 4/5. Since 36 is larger than 35, 4/5 > 7/9
Never split the denominator
The numerator may be split, but never split the denominator.
Comparing Fractions using Benchmark Values
Estimate values using benchmarks. For example: What is 10/22 of 5/18 of 2000?If you recognize that 10/22 is nearly 1/2 and 5/18 is approx. 1/4 then it is easier to determine. Try to make rounding errors cancel by rounding some numbers up and other numbers down
Unspecified Number Amounts
Use Smart numbers. To make computation easier, choose numbers equal to common multiples of the denominators of the fraction in the problem.
Percent problems
The key is to make them concrete by picking real numbers with which to work
Percents as Fractions
Part/Whole = Percent/100Fill in the table, set up as a proportion, cancel cross-multiply and solve
Benchmark Values: 10%
To find 10% of any number, just move the decimal point to the left one place
Percent Increase and decrease
Use the percent table however adjust it for change instead of part. Change/Original = Percent/100Also can do so with following equations:ORIGINAL x (1+% increase/100) = NEWORIGINAL x (1-% decrease/100) = NEW
Successive Percents
Best to solve by choosing real numbers and seeing what happens-- 100 is usually the easiest number
Chemical Mixtures
Set up a mixture chart with the substance labels in rows and "original," "change" and "new" in the columns. This way you can keep track of various components and their changes.
When to use Fractions vs. decimals
Fractions are good for cancelling factors in multiplications or expressing proportions that do not have clean decimal equivalents. Decimals are good for estimating results or for comparing sizes.Prefer fractions for doing multiplication/division but prefer decimals and percents for doing addition or subtraction, estimating numbers or comparing numbers
Estimating Decimal Equivalents
1) Make the denominator the nearest factor of 100 or another power of 10.2) Change the numerator or denominator to make the fraction simplify easily. 3) Make percent adjustments by seeing how much you approximately changed the denominator and applying that percent to the final answer.
Data Sufficiency strategy
Rephrasing: You should be able to rephrase the equation to have one equation with one variable. If you are unable to do this, the stem is not sufficient.
Percent shortcuts
When trying to find a more complicated percentage, break it into easy to find chunks.For example: 23% of 400: 10% of 400 is 40 therefore 20% is 2 x 40 = 80. 1% of 400 is 4 and 3% is 3 x 4 or 12. Putting it together, we get 80+12=92
When to try plugging in numbers
1) variables in the answer choices2) percents in the answer choices (when they are percents of some unspecified amount)3)Fractions or ratios in teh answer choices (when they are fractional parts or ratios of unspecified amounts)
Quadrilaterals
four sided shapestrapezoids, parallelograms and special parallelograms such as rhombuses, rectangles and squares
Parallelogram
Opposite sides and opposite angles are equal
Rectangle
All angles are 90 degrees and opposite sides are equal
Square
All angles are 90 degrees and all sides are equal
Rhombus
All sides are equal. Opposite angles are equal.
Trapezoid
One pair of opposite sides is parallel
Polygons and Interior Angles
Sum of Interior Angles of a polygon = (n-2) x 180Triangle has 3 sides and 180 degreesQuadrilateral has 4 sides and 360 degreesPentagon has 5 sides and 540 degreesHexagon has 6 sides and 720 degreesAnother way to find the sum is to divide the polygon into triangles
Polygons and perimeter
Perimeter is the distance around a polygon and equals the sum of all sides
Polygons and Area
Area refers to the space inside the polygon measured in square units
Area of a Triangle
(Base x Height) / 2
Area of Rectangle
Length x Width
Area of Trapezoid
[(Base 1 + Base 2) x Height] / 2Average of two bases multiplied by the height
Area of Parallelogram
Base x Height
Area of rhombus
(Diagonal 1 x Diagonal 2) / 2Diagonals are always perpendicular bisectors (cut each other in half at a 90 degree angle) in a rhombus
Surface Area
Find the sum of all the faces. Find the area of each face
Volume
the amount of stuff a shape can holdVolume = Length x Width x Heightessentially it is equal to the area of the base multiplied by the heightRemember when fitting 3D objects into other 3D opjects, knowing the volumes is not enough. We must know the specific dimensions (length, width and height) of the object to determine whether it will fit
Angles of a Triangle
(1) The sum of the three angles of a triangle equals 180 degrees(2) Angles correspond to their opposite sides. The largest angle is opposite the longest side and vice versa. If two sides are equal, their opposite angles are also equal (isosceles).
Sides of a triangle
The sum of any two sides of a triangle must be GREATER THAN the third side. If you are given two sides of a triangle, the length of the third side must lie between the difference and the sum of the two given sides.
Pythagorean Theorem
A right triangle (with a 90 degree angle) is composed of two legs and a hypotenuse (side opposite the right angle). The sum of the square of two legs equals the square of the hypotenuse. a^2 + b^2 = c^2
Common Right Triangles: 3-4-5
Key multiples: 6-8-109-12-1512-16-20
Common Right Triangles: 5-12-13
10-24-26
Common Right Triangles: 8-15-17
...
Isosceles Triangles
Most popular is the 90 degree:45-45-901:1: sq rt. 2x:x:x sq rt 2Important bc it is half of a square
Equilateral triangles
All three sides equal and therefore all angles equal to 60 degrees
30-60-90
often formed from equilateral triangles1: sq rt 3:2
Delux Pythagorean Theorem
d^2=x^2+y^2+z^2where x, y and z are the sides of the rectangular solid and d is the main diagonal
Similar Triangles
triangles with equal corresponding angles and proportional corresponding sides. If 2 triangles have 2 pairs of equal angles you know they are similar triangles. If two similar triangles have corresponding side lengths in ratio of a:b, then their areas will be in ratio a^2:b^2--this holds true for any similar figures. For similar solids with corresponding sides in ratio a:b, their volumes will be in ratio a^3:b^3
Equilateral Triangle
Can be split ito two 30-60-90 triangles. Therefore it has a base of length S and a height of length S sq.rt.3 / 2. Area of an equilateral triangle therefore is S^2 sq.rt.3 / 4
Circles & cylinders
Circle is a set of points in a plane that are equidistant from a fixed center point. A line segment connecting the center point to a point on the circle is the radius. Any line segment connecting two points on a circle is a chord. Any chord that passes through the center of the circle is a diameter. A diameter is two times the radius. GMAT tests:(1) circumference and (2) area of whole and partial circles, (3) surface area and (4) volume of cylinders
Circumference of a circle
The distance around a circle. C=2 pi rpi= approx 3.14C=d piA full revolution of a wheel is equivalent to the wheel going around once
Circumference and arc length
an arc is a portion of the circle rather than the whole. Arc length can be determined by what fraction the arc is of the entire circumference
Perimeter of a sector
The boundaries of a sector of a circle are formed by the arc and two radii (like a slice of pizza--the arc is the crust and the center of the circle is the tip of the slice). If you know the length of the radius and the central angle, you can find the perimeter of the sector
Area of a circle
A= pi r^2
Area of a sector
Find the fraction of the circle the sector represents and multiply by the area of the circle.
Inscribed vs. central angles
An inscribed angle has a vertex on the circle itself. An inscribed angle is equal to half of the arc it intercepts
Inscribed triangles
If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be a right triangle. Vice versa.
Cylinders and Surface Area
A cylinder is two circles and a wrapped up rectangle. The length of the rectangle is equal to the circumference of the circle (2pi r) and the width of the rectangle is equal to the height of the cylinder, h. Area of the rectangle is 2pi r x hSurface Area= 2 circles + rectangleSA= 2( pi r^2) + 2pi rhThe only information needed to find SA is the radius of the cylinder and the height.
Cylinders and volume
V=pi r^2hOnly need the radius of the cylinder and height of the cylinderTwo cylinders can have the same volume but different shapes
Properties of Intersecting Lines
(1) Interior angles formed by intersecting lines form a circle so the su, of the angles is 360 degrees(2) Interior angles that combine to form a line sum to 180 degrees(3) Angles found opposite each other where these two lines intersect are equal--vertical angles
Exterior angles of a triangle
an exterior angle of a triangle is equal in measure to the sum of the two non-adjacent (opposite) interior angles of the triangle. Tested frequently on the GMATIn particular look for exterior angles within more complicated diagrams. Perhaps try isolating the triangle.
Parallel lines cut by a transversal
GMAT makes frequent use of diagrams that include parallel lines cut by a transversalAll acute angles (less than 90 degrees) are equalAll obtuse angles (greater than 90 degrees but less than 180 degrees) are equalAlways be on the look out for parallel lines and extend lines and label the acute and obtuse angles. You might also label the parallel lines with arrows.
Slope of a line
rise/run or change in y/change in x
Types of slopes
Think of slope as walking from left to right. If you walk along a line with a positive slope, you would walk up. A horizonal line has a zero slope and a vertical line has an undefined slope
Intercepts of a line
a point where a line interescts a coordinate axis is called an intercept. X-intercept is where the line intersects the x-axis and y-intercept is where the line crosses the y-axis. The x-intercept is the point on the line in which y=0The y-intercept is the point on the line in which x=0. To find the x-intercept, plug in 0 for y. Do the same technique to find the y-intercept.
Slope-intercept equations
y=mx+bm represents the slope of the line and b represents the y-intercept of the line. Vertical lines take the form x=some numberHorizontal lines take the form y=some numberLinear equations take the shape Ax+By=C and never have sq. roots, squares or xy.
Step by Step from 2 points on a line
1: Find slope by calculating rise over run (if only one point and y-int are given, use y-int as a point (0,y)2: Plug the slop into the equation3: Use the given point to plug in for the values of y and x to find b4: Write the complete equation
Distance between 2 points
Calculate using the pythagorean theorem. Draw a right triangle between the two points.
Perpendicular bisectors of line segments
Forms a 90 degree angle with the segment and divides the segment in half. Has a negative reciprocal slope. In order to find a point on the perpendicular bisector, remember that it passes through the midpoint of the original line segment. Thus solve for the midpoint and that will provide you a point on the line of the perpendicular bisector. Use the midpoint as a point on the line to plug into the equation in order to find the y-intercept.
Intersection of Two Lines
If two lines in a plane intersect in a single point, the coordinates of that point solve the equations of BOTH lines. If two lines in a plane do not intersect, then the lines are parallel and there is no pair of numbers that satisfies both equations at the same time.
Maximum Area of Quadrilateral
Question may be asked explicitly or implicitly (such as Is the area of rectangle ABCD less than 30?)Typically maximizing the area of a quadrilateral (usually a rectangle) with a fixed perimeter. Of all quadrilaterals with a given perimeter, the SQUARE has the largest area. Of all quadrilaterals with a given area, the SQUARE has the minimum perimeter.
Maximum area of parallelogram or triangle
Another common optimization problem involves maximizing the area of a triangle or parallelogram with given side lengths.If you are given two sides of a triangle or parallelogram and you want to maximize the area, establish those sides as the base and height and make the angle between them 90 degrees. In other words, if you are given two sides of a triangle or parallelogram, you can maximize the area by placing those two sides perpendicular to each other.
Function Graphs and Quadratics
Quadratic functions look like:ax^2 +bx+c and take the form of a parabola. Positive value of a --> parabola curves upwardNegative value of a--> parabola curves downwardLarge abs. value of a-->narrow curveSmall abs. value of a-->wide curveMost likely questions asked are how many x-intercepts and what are they?The parabola touches the x-axis at those values of x that make f(x)=0Sometimes you have to use the quadratic formula. If you do, the discriminant (b^2-4ac) located under the radical sign will tell you how many solutions the equation has.
What the discriminant tells you
(1) If positive-->sq. root yields a positie number and it produces 2 roots (2 x-intercepts)(2) If it equals zero, sq. root yields zero. Only produces 1 root and the parabola has just 1 x-intercept(3) If negative-->sq. root cannot be performed and produces no roots, no x-intercepts
Simultaneous Equations: solving by substitution
Use substitution whenever one variable can be easily expressed in terms of the other.
Simultaneous Equations: solving by combination
Add or subtract the two equations to eliminate one of the variables. Use whenever it is easy to manipulate the equation so that the coefficients for one variable are the SAME or OPPOSITE.
Mismatch Problems
Just because there are 3 variables does NOT mean 3 equations are necessary. For example, if they are only asking for one variable it may be possible to determine from only 2 equations. Follow through with the algebra to determine if the solution can be found. Be on the look out for exponents. 2 variables with 2 equations may not be solvable if there are exponents present. MASTER RULE for determining whether 2 equations involving 2 variables will be sufficient to solve:(1) If both of the equations are linear and there are no squared terms or xy terms, the equations will be sufficient unless two equations are mathematically identical(2) If there are any non-linear terms there will USUALLY be two or more different solutions for each of the variables and the equations will not be sufficient.
Combo problems: manipulations
GMAT often asks to solve for a combination of variables, for example: x+yIn these cases do no isolate and solve for the individual variables until all other methods have been exhausted. Instead try to isolate the combination on one side. 4 manipulations (MADS):M: Multiply or divide the whole equation by a certain numberA: Add or subtract a number on both sides of the equationD: Distribute or factor an expression on ONE side of the equationS: square or unsquare both sides of the equationOccur most frequently in data sufficiency:manipulate the equation so that the combo is one one side. If a value is on the other side, it's sufficient. If a variable expression is on the other side it is not sufficient.
Absolute value equations
(1) Isolate the expression within the absolute value brackets(2) Remove the absolute value brackets and solve for the equation in 2 cases. Case 1: x=a (x is positive). Case 2: x=-a (x is negative)(3) Check to see whether each solution is valid by putting each one back into the original equation and verifying that the two sides of the equation are in fact equal.
Even exponent equations: 2 solutions
Even exponents hide the sign of the base and therefore can have positive and negative solutions. This is often the case even for equations with some odd exponents and some even exponents.
Same base or same exponent
In problems that involve exponential expressions on BOTH sides or the equation, it is best to REWRITE the bases so that either the same base or the same exponent appears on both sides of the exponential equation. Be careful however if 0,1, or -1 is the base since the outcome of raising those bases to powers is not unique.
Eliminating Roots: Square both sides
To solve variable square roots, square both sides of the equation. Be sure to check the solution in the original equation since squaring can introduce an extraneous solution.
Factoring: three Special Products
x^2-y^2 =(x+y)(x-y)x^2+2xy+y^2= (x+y)(x+y) = (x+y)^2x^2-2xy+y^2 = (x-y)(x-y) = (x-y)^2
Defining rules for Sequences
Linear sequences are in the form kn+x, where k equals the difference between successive terms. Each term is equal to the previous term plus a constant k. Exponential sequences are in the form of x(k^n) where x and k are real numbers. Each term is equal to the previous term times a constant k.
Sequences and Rules
Sometimes a rule is too difficult, one can also look for patterns and apply those patterns.
Direct Proportion Functionality Types
Set up ratios fir the before and after cases and then set the ratios equal to each other