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GMAT: Equations, Inequalities, & VICs
Manhattan Prep Book 3
Terms in this set (54)
1. Substitution: solve for 1 variable and substitute it in.
2. Combination: add/subtract the two equations to eliminate one of the variables. Bk. 3, pg. 15
3. Three equations: use substitution or combination.
To determine if 2 equations with two variables will be sufficient to solve for 1 value:
1. Both of the equations are linear (no exponents), no xy terms, not mathematically identical.
2. Cannot solve for 1 solution if any non-linear terms
Try not to solve for individual variables right away.
M: multiply or divide the whole equation by a certain number.
A: add or subtract a number on both sides of the equation.
D: distribute or factor an expression on ONE side of the equation.
S: square or un-square both sides of the equation.
Isolate the combo on one side of the equation.
Combos in Data Sufficiency
Manipulate the given equations so that the combo (variables) is isolated on one side of the equation.
A: if the other side of the equation is a VALUE, that expression is SUFFICIENT.
B: if the other side of the equation is a VARIABLE, the equation is NOT sufficient
Absolute Value Equations
Generally have TWO solutions.
1. Isolate the absolute value expression on one side.
2. Remove the absolute value brackets and solve for x=a, x=-a.
3. Check whether each solution is valid by inputting in original equation.
- sometimes a solution may fail!
EVEN: dangerous! Hide the sign of the base. Can have both a positive and negative solution (like absolute value equations).
ODD: have only one solution.
If a mix of odd and even exponents, likely to have 2 solutions.
Must rewrite the bases so that either the same base or same exponent appears on both sides of the equation.
Be careful of bases 0, 1, & -1 as the outcomes are not unique.
Equations with Square Roots
Square both sides of the equation.
Check that the solution(s) work in the original equation.
Square root symbols only work over positive numbers.
Have one unknown and two defining components:
1. A variable raised to the 2nd power
2. A variable raised to the 1st power
Generally have two solutions.
Factoring Quadratic Equations
1. Move all terms to one side of the equation, combine them and put in standard form. Set other side to 0.
2. Factor the equation. Find two integers whose product equals C and whose sum equals B.
3. Rewrite the equation as (x + ?)(x + ?) using the numbers found in 2.
4. Set each factor to 0 and solve for x.
Some Quadratics are hidden within equations.
If you have a quadratic expression equal to 0, AND you can factor an x out of the expression, then x=0 is one solution.
Be careful not to divide both sides by x.
One-solution Quadratics are perfect square quadratics. Both roots are the same.
Zero can never be in the denominator of an expression.
Three special products: study book 3, pg. 48
Formula is given and solve for one of the variables by plugging in given values.
Tip: write the formula as part of an equation.
Trick: formulas maybe unfamiliar.
Strange Symbol Formulas
An arbitrary symbol which defines a certain procedure.
- carefully follow each step in the procedure that the symbol indicates.
1. Refer to the variables as the 1st number, 2nd number, etc.
2. Calculate a solution with actual numbers.
3. Always perform procedures in parentheses first.
Tricks: watch for symbols that invert the order of operations.
Formulas with Unspecified Amounts
These questions focus on the increase or decrease in the value of the formula, given a change in the value of variables.
Tip: pick smart numbers.
A collection of numbers in a set order. The order is determined by a RULE.
Each item of the sequence is defined as a function of N, the place in which the term occurs in the sequence.
Rules for Sequences
You MUST be given the rule in order to find the number in a sequence. You cannot solve a sequence without a rule.
Linear Sequence: difference between terms is always the same.
Exponential Sequence: each term equals the previous term times a constant.
Sequence Problems: Alternate Methods
For simple linear sequences, instead of following the rule, find how many "jumps" there are and multiply that by the constant. Then add the product to the first term to find the needed term.
Book 3, pg. 61
Some sequences are easier to look at in terms of patterns rather than rules.
A function rule describes a series of operations to perform on a variable.
- never possible to generalize a rule only by using specific cases.
Domain of a function: the possible inputs
Range of a function: the possible outputs
Numerical: input a numerical value in place of the independent variable to determine f(x).
Variable: input an algebraic expression in place of the independent variable and then simplify to find f(x).
- keep the expression in parentheses
A series of functions where the output from the inner function is the input for the outer function.
Tip: changing the order of the functions changes the answer.
Tip: if asked to find a value of x for which f(g(x)) = g(f(x)), use variable substitution
Functions with Unknown Constants
Functions may have an unknown constant:
1. Use the given input/output variable to solve for the unknown constant.
2. Rewrite the function using the constant found in 1.
3. Solve the function for the new input variable.
Visualize functions by graphing:
Input = x-coordinate (domain)
Output = y-coordinate (range)
Plot points of inputs/outputs on a coordinate plane.
Common Functions: Proportionality
Direct: two quantities always change by the same factor and in the same direction.
y=kx; x is input, y is output, k is proportionality constant
y/x = k
From given 'before' and 'after' values, set up ratios to solve
Inverse: two quantities change by reciprocal factors
Y=k/x; x is input, y is output, k is proportionality constant
Set up products of x & y to solve.
Common Functions: Linear Growth
Quantities that grow or decay at a constant rate.
y = mx + b; m is constant rate, y is b value at time 0, x is time
Compares quantities that have different values.
Use number lines to help visualize.
Can solve like equations EXCEPT when multiplying or dividing an inequality by a negative number. The sign will FLIP.
- cannot multiply or divide by a variable unless you know the sign of the number the variable stands for.
Series of inequalities strung together.
1. Solve any inequalities that need to be solved.
2. Simplify inequalities so all inequality symbols point in the same direction (preferably left <).
3. Line up common variables in the inequalities.
4. Combine inequalities by taking the more limiting upper and lower extremes.
Not always possible to combine all inequalities when working with variables.
Manipulating Compound Inequalities
If you perform operations on a compound inequality, remember to perform those operations on EVERY term in the inequality in the same way.
Combine by adding them together. Make sure the inequality signs are facing the same direction.
NEVER subtract or divide two inequalities together.
ONLY multiply when both sides of both inequalities are positive.
Using Extreme Values
Focus on the extreme values of an inequality when
1. Problems with multiple inequalities where the question involves the potential range of values for variables in the problem.
- plug extreme values using LT for less than or GT for greater than
2. Problems involving both equations and inequalities.
- substitute values from one into the other.
- first simplify the inequality
- plug the extreme values from the inequality into the equation
- may have to change the extreme value sign based on the operations
Book 3, pg. 92
Focus on the largest and smallest possible values for each of the variables to find the largest or smallest possible result.
Make tables to organize variables/solutions; check the endpoints of the ranges for each variable.
Squared terms are minimized when set equal to zero.
Testing Inequality Cases
Test positive/negative cases for variables in an inequality.
Use positive/negative rules to set up scenarios/tables for many inequality problems.
Consider testing positive/negative when there is a zero on one side of the inequality.
Book 3, pg. 95
Inequalities & Absolute Value
Try to visualize the problem with a number line.
Inequality/absolute value equations generally have more than two possible solutions.
Center of graph is -b; x must be exactly c units away from -b.
Algebraically: solve 2 times. Simply remove brackets & remove brackets and change sign
Inequalities involving even exponents have 2 scenarios.
When you take the square root of an inequality, you may need to evaluate an absolute value scenario.
ONLY take the square root of an inequality for which both sides are definitely NOT negative.
Book 3, pg. 99
Have VARIABLE expressions IN the answer CHOICES.
- are always problem solving problems.
- always test some OTHER math content.
- can generally be solved in more than one way.
- some require you to invent variables to solve algebraically.
- for some you cannot pick numbers for every variable in the problem.
Types of VICs
1. Word Translations: word problems where algebraic equations solve for the remaining unknown.
2. Algebraic: variables don't represent anything; use algebra to isolate the expression asked about and whatever is on the other side is the answer.
3. Percent: interpret statements and devise equation(s) matching the problem.
4. Geometry: draw a diagram and use geometry to solve.
Strategies for Solving VICs
1. Direct Alegbra: see the correct algebraic translation of the problem.
2. Pick Numbers & Calculate a Target: plug numbers into answer choices to see if they yield the same value as the target.
- could pick #s resulting in 2 choices and time consuming.
3. Hybrid Method: pick #s to think through the computations step by step.
VICs: Picking Numbers
It can be helpful to use a chart. Test EVERY answer choice.
1. Never pick 1 or 0 and avoid 100 on percent problems.
2. Make sure all numbers are different.
3. Pick small numbers.
4. Try to pick prime numbers.
5. Avoid numbers that appear as a coefficient in several answer choices.
VICs: Direct Algebra
Break the problem down into manageable parts.
Think through the algebra step by step, writing down expressions for intermediate quantities on EVERY step.
VICs: Hybrid Method
Pick numbers to help think through the problem.
Break the problem into manageable parts.
Think through step by step using the numbers and related algebraic expressions.
Which VIC Strategy Do I Use?
Some problems are much easier to solve using one technique over another.
Practice each technique as some are faster.
When in doubt, pick numbers and find a target.
- if stuck on a problem change to picking numbers.
- a VIC can only be solved by direct algebra if variables are already defined as numbers.
Complex Absolute Value Equations
1. The equation contains 2 or more variables in more than one absolute value expression. A conceptual approach is preferred.
2. The equation contains 1 variable and at least one constant in more than one absolute value expression.
- easier to solve algebraically.
- only consider one case in which neither expression changes sign and another where only one expression changes sign. Check answers!
There may be many possible solutions among ALL numbers but only one INTEGER solution.
Solve for one variable and then test numbers.
Integer constraints together with inequalities can also lead to just one solution.
Multiplying/Dividing Two Equations
Apply this technique when you can cancel a lot of variables at once.
Multiply the left sides of the equation together and multiply the right sides of the equations together. Set the products equal to each other and solve. To divide, follow the same steps.
Advanced Factoring and Distributing
Whenever there is an expression or equation in which two or more terms include the same variable, consider factoring as an approach. When given in factored form, consider distributing.
Advanced Quadratic Techniques
Square Roots: if one side of a quadratic is a perfect square, the problem can be solved by taking the square root of both sides. Make sure to take the absolute value to get both positive and negative solutions.
Substituting: substitute a variable to create a quadratic. When the quadratic is solved with the substitution, use those answers to solve for the original variable.
Quadratic Formula: for any equation ax^2 + bx + c = 0,
X = (-b+\-(Sq. Root(b^2 - 4ac)))/2a
Expression underneath the radical is the discriminate:
Discriminate > 0, two solutions
Discriminate = 0, one solution
Discriminate < 0, no solutions
Each item is defined in terms of the value of PREVIOUS items in the sequence.
- need to know at least 1 term to solve.
Linear (arithmetic): difference between terms is always the same.
Exponential (geometric): ratio between terms is always the same.
Solve for constants k & x and write the direct formula for the missing term.
A quantity is multiplied by the same constant each period of time.
Exponential growth multipliers commonly take the form of percentage multipliers.
Two seemingly different inputs to the function always yield the same output.
1. Substitute the functions given into the answer choices and solve algebraically to see if they give the same result.
2. Pick a number for x and see which answer choice gives the desired results.
Property problems: simply pick numbers.
Optimization: Linear Problems
Have straight-line graphs.
The extreme (min & max) occur at the boundaries - at the smallest possible x and the largest possible x as given in the problem.
Optimization: Quadratic Functions
Form parabolas: have a peak (max) or a valley (min).
Key: make the squared expression equal to 0. Whatever x makes the squared expression 0 is the min/max value of x. The resulting y is the min/max value of the function.
When you have to multiply or divide by a variable before you know the sign of the variable, set up two cases (pos. & neg.) and solve each separately to get two scenarios.
*be careful on data sufficiency questions.
Reciprocals of Inequalities
You need to consider the positive/negative cases of the variables involved. You cannot take the reciprocal unless you know the signs.
You should flip the inequality unless x & y have different signs.
1/x > 1/y when both are positive
1/x > 1/y when both are negative
1/x < 1/y when x is neg. and y is pos.
You cannot square both sides of an inequality unless you know the signs of both sides.
- if both sides are negative, flip the sign.
- if both sides are positive, don't flip the sign.
- if signs are 1 negative & 1 positive, then you cannot square.
- if the signs are unclear, you cannot square.