Gmat Math 03 - Equations , Inequalities , Vics

Front 1

If you plan to add the equations, multiply one or both of the equations so that the coefficient of a variable in one equation is the ... of that variable's coefficient in the other equation.

Back 1

OPPOSITE

Front 2

If you plan to subtract them, multiply one or both of the equations so that the coefficient of a variable in one equation is the ... as that variable's coefficient in the other equation.

Back 2

SAME

Front 3

The procedure for solving a system of three equations with three variables is exactly the same as for a system with two equations and two variables. You can use ...

Back 3

Substitution or combination

Front 4

What is the sum of x, y and z?

x+y=8
x+z=11
y+z=7

Back 4

In this case, DO NOT try to solve for x, y, and z individually. Instead, notice the symmetry of the equations-each one adds exactly two of the variables-and add them all together:
x + y = 8
x + z = 11
+ + y+ z = 7
______________
2x+ 2y+ 2z = 26

Therefore, x +y + z is half of 26, or 13.

Front 5

Mismatch problem: What is x?

(1) (3x) / (3y+5z) = 8
(2) 6y + 10z = 18

Back 5

It is tempting to say that these two equations are not sufficient to solve for x, since there are 3 variables and only 2 equations. However, the question does NOT ask you to solve for all three variables. It only asks you to solve for x, which IS possible:
First, get the x term on one side of the equation:
Then, notice that the second equation gives us a value for 3y + 5z, which we can substitute into the first equation in order to solve for x:
3x / (3y+ 5z) = 8
3x = 8(3y + 5z)
6y + 10z = 18
2(3y + 5z) = 18
3y+ 5z= 9
3x = 8(3y + 5z)
3x= 8(9)
x= 8(3) = 24
The answer is (C): BOTH statements TOGETHER are sufficient

Front 6

What is x?

(1) y = x^3-1
(2) y=x-1

Back 6

It is tempting to say that these 2 equations are surely sufficient to solve for x, since there are 2 different equations and only 2 variables. However, notice that if we take the expression for y in the first equation and substitute into the second, we actually get multiple possibilities for x.
x^3 -1 = x-1
x(x^2-1)=0
x(x + 1)(x-1)=0
x^3-x=0
x= {-1,0,1}
Because of the exponent (3) on x, it turns out that we have THREE possible values for x. If x equals either -1, 0, or 1, then the equation x^3 = x will be true. We can say that this equation has three solutions or three roots. Therefore, we cannot find a single value for x. The answer to the problem is (E): the statements together are NOT sufficient.

Front 7

A MASTER RULE for determining whether 2 equations involving 2 variables (say, x and y) will be sufficient to solve for the variables is this:

Back 7

1) If both of the equations are linear-that is, if there are no squared terms (such as x^2 or y^2) and no xy terms - the equations will be sufficient UNLESS the two equations are mathematically identical (e.g., x +y = 10 is identical to 2x + 2y = 20).
(2) If there are ANY non-linear terms in either of the equations (such as x^2, y^2, xy, or x/y), there will USUALLY be two (or more) different solutions for each of the variables and the equations will not be sufficient.

Front 8

There are four easy manipulations that are the key to solving most COMBO problems. You can use the acronym ... to remember them.

Back 8

MADS
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 unsquare both sides of the equation.

Front 9

Equations that involve absolute value generally have ... SOLUTIONS.

Back 9

TWO

Front 10

x^2 + 9 = 0 ??

Back 10

x^2= -9 ??

Squaring can never produce a negative number!
This equation does not have any solutions.

Front 11

Quadratic equations are equations with one unknown and two defining components:

Back 11

(1) a variable term raised to the second power
(2) a variable term raised to the first power

Front 12

In the equation x^2 + 3x - 4 = 0, we can see that b = 3 and c = -4. In order to factor this equation, we need to find two integers whose product is -4 and whose sum is 3 ...

Back 12

The only two integers that work are 4 and -1, since we can see that 4(-1) = -4 and 4 + (-1) = 3.

Front 13

If you have a quadratic expression equal to 0, and you can factor an x out of the expression, then ... is a solution of the equation.

Back 13

x = 0. Be careful not to just divide both sides by x.

Front 14

One-solution quadratics are also called ...

Back 14

Perfect square quadratics. Be careful not to assume that a quadratic equation always has two solutions. Always factor quadratic equations to determine their solutions. In doing so, you will see whether a quadratic equation has one or two solutions.

Front 15

When ... appears in the denominator of an
expression, then that expression is undefined.

Back 15

zero

Front 16

If either of the factors in the ... is 0, then the entire expression becomes 0

Back 16

Numerator

Front 17

Special Product #1: x^2-y^2=

Back 17

(x+y)(x-y)

Front 18

Special Product #2: x^2 + 2xy +y^2 =

Back 18

(x +y)(x +y) = (x + y)^2

Front 19

Special Product #3: x^2 - 2xy +y^2 = (x - y)(x - y) =

Back 19

(x - y)^2

Front 20

An important thing to remember about sequence problems is that you MUST be given the ... in order to find a particular number in a sequence.

Back 20

Rule. It is tempting (but incorrect!) to try to solve sequence problems without the rule.

Front 21

A common sequence definition is a linear (or arithmetic) sequence. In these sequences, the
difference between successive terms is always the same. The direct definition of a linear sequence is ...

Back 21

Sn = kn + x, where k is the constant difference between successive terms and x is some other constant.

Front 22

The first four terms of a sequence are 16, 20, 24, and 28, in that order. If the difference between each successive term is constant, what is the rule for this sequence?

Back 22

Because we know that the difference between successive terms is the same, we know the sequence is linear. The difference between each term is 4, so this sequence must be in the form 4n + x. We can find the value of x by using any of the terms in the sequence. In this example, the first term in the sequence (n = 1) has a value of 16, so 4(1) + x = 16.
Therefore, x = 12.
We can confirm that x = 12 by using another term in the sequence. For example, the second term in the sequence (n = 2) has a value of 20. This verifies that x = 12, since 4(2) + 12 = 20. Therefore, 4n + 12 is the rule for the sequence. We can use this formula to find
any term in the sequence. 55 = 4*5 +12 = 32; 56 = 4*6 +12 = 36; etc.
Another common sequence definition is an exponential (or geometric) sequence. These
sequences are of the form S; = x(kn), where x and k are real numbers. Each term is equal to the previous term times a constant k. (In contrast, in a linear sequence, each term is equal to the previous term plus a constant k, as we saw earlier)

Front 23

Another common sequence definition is an exponential (or geometric) sequence. These sequences are of the form ...

Back 23

Sn = x(k^n), where x and k are real numbers. Each term is equal to the previous term times a constant k. In contrast, in a linear sequence, each term is equal to the previous term plus a constant k, as we saw earlier.

Front 24

The first four terms of a sequence are 20, 200, 2,000, and 20,000, in that order. If each term is equal to the previous term times a constant number, what is the rule for this sequence?

Back 24

Because we know that each term is equal to the previous term times a constant number, we know the sequence is exponential. Each term is 10 times the previous term, so this sequence must be in the form x(10^n). We can find the value of x by using any of the terms in the sequence. In this example, the first term in the sequence (n = 1) has a value of 20, so x(10^1)
=20.Therefore, x=2. S1=2*10^1=20; S2=2*10^2=200; etc.

Front 25

You put a 2 into the magic box, and a 7 comes out. You put a 3 into the magic box, and a 9 comes out. You put a 4 into the magic box, and an 11 comes out. What is the magic box doing to your number?

Back 25

There are many possible ways to describe what the magic box is doing to your number.
One possibility is as follows: The magic box is doubling your number and adding 3.
2(2) + 3 = 7 2(3) + 3 = 9 2(4) + 3 = 11
Assuming that this is the case (it is possible that the magic box is actually doing something different to your number), this description would yield the following "rule" for this magic box: 2x +"3. This rule can be written in function form as:
f(x) = 2x+ 3.

Front 26

By the way, the expression f(x) is pronounced ...

Back 26

" f of x", not "fx." It does NOT mean "f TIMES x"! The letter f does NOT stand for a variable.

Front 27

Direct proportionality means that the two quantities always change by the same factor and in the same direction. For instance, tripling the input will cause the output to triple as well.
Cutting the input in half will also cut the output in half. Direct proportionality relationships are of the form ...

Back 27

Y = kx, where x is the input value and Y is the output value. k is called the proportionality constant. This equation can also be written as y/x= k, which means
that the ratio of the output and input values is always constant.

Front 28

The maximum height reached by an object thrown directly upward is directly proportional to the square of the velocity with which the object is thrown.
If an object thrown upward at 16 feet per second reaches a maximum height of 4 feet, with what speed must the object be thrown upward to reach a maximum height of 9 feet?

Back 28

Typically with direct proportion problems, you will be given "before" and "after" values.
Simply set up ratios to solve the problem-for example, y1/x1 can be used for the "before" values and y2/x2 can be used for the "after" values. We then write y1/x1 = y2/x2 , since both ratios are equal to the same constant k. Finally, we solve for the unknowns.
In the problem given above, be sure to note that the direct proportion is between the height
and the square of the velocity, not the velocity itself Therefore, write the proportion as h1/(v1^2) = h2/(v2^2). Substitute the known values h1 = 4, v1= 16, and h2 = 9:

4/(16^2)=9/(v^2)
v2^2=9*((16^2)/4
v2^2=9*64=576
v2=24
The object must be thrown upward at 24 feet per second.

Front 29

Inverse proportionality means that the two quantities change by ... factors.

Back 29

RECIPROCAL. Cutting the input in half will actually double the output. Tripling the input will cut the output to one-third of its original value.

Front 30

Inverse proportionality relationships are of the form ...

Back 30

y=k/x, where x is the input value and y is the output value. k is called the proportionality constant. This equation can also be written as xy = k, which means that the product of the output and input values is always constant.

Front 31

The amount of electrical current that flows through a wire is inversely proportional to the resistance in that wire. If a wire currently carries 4 amperes of
electrical current, but the resistance is then cut to one-third of its original value, how many amperes of electrical current will flow through the wire?

Back 31

While we are not given precise amounts for the "before" or "after" resistance in the wire, we can pick numbers. Using 3 as the original resistance and 1 as the new resistance, we can see that the new electrical current will be 12 amperes:
C1R1=C2R2
4(3)=C2(1)
12=C2

Front 32

linear growth (or decay)-that is, they grow at a constant rate. Such quantities are determined by the linear function:

Back 32

Y = mx + b. In this equation, the slope m is the constant rate at which the quantity grows.

Front 33

Jake was 4.5 feet tall on his 12th birthday, when he began to have a growth spurt. Between his 12th and 15th birthdays, he grew at a constant rate. If Jake
was 20% taller on his 15th birthday than on his 13th birthday, how many inches per year did Jake grow during his growth spurt? (12 inches = 1 foot)

Back 33

In this problem, the constant growth does not begin until Jake has reached his twelfth birthday, so in order to use the constant growth function y = mx + b, let time x = 0 (the initial state) stand for Jake's twelfth birthday. Therefore, x = 1 stands for his 13th birthday, x =2 stands for his 14th birthday, and x = 3 stands for his 15th birthday.
The problem asks for an answer in inches but gives you information in feet. Therefore, it is convenient to convert to inches at the beginning of the problem: 4.5 feet = 54 inches = b.
Since the growth rate m is unknown, the growth function can be written as y == mx + 54.
Jake's height on his 13th birthday when x = 1, was 54 + m; and his height on his 15th birthday, when x = 3, was 54 + 3m, which is 20% more than 54 + m. Thus, we have:
54 + 3m = (54 + m) + 0.20(54 + m)
54 + 3m = 1.2(54 + m)
54 + 3m = 64.8 + 1.2m
1.8m = 10.8
m = 6
Therefore, Jake grew at a rate of 6 inches each year.

Front 34

When you multiply or divide an inequality by a negative number, the ... flips!

Back 34

Inequality sign

Front 35

you cannot multiply or divide an inequality by a variable, unless you know the ... of the number that the variable stands for. The reason is that
you would not know whether to flip the inequality sign.

Back 35

Sign

Front 36

You can perform operations on a compound
inequality as long as you remember to perform those operations ...

Back 36

On every term in the inequality, not just the outside terms

Front 37

As discussed above, many GMAT inequality problems involve more than one inequality. Another helpful approach is ...

Back 37

To combine inequalities by adding the inequalities together.
In order to add inequalities, we must make sure the inequality signs are facing the same direction.

Front 38

Is mn < 10?
(1) m < 2
(2) n < 5

Back 38

It is tempting to multiply these two statements together and conclude that mn < 10. That would be a mistake, however, because both m and n could be negative numbers that yield a number larger than 10 when multiplied together. For example, if m = -2 and n = -6, then mn = 12, which is greater than 10.
Since you can find cases with mn < 10 and cases with mn > 10, the correcr answer is (E):
The two statements together are INSUFFICIENT to answer the question definitively.

Front 39

If m and n are both positive, is mn < 107

Back 39

Since the variables are positive, we can multiply these inequalities together and conclude that mn < 10. The correct answer is (C).
Only multiply inequalities together if both sides of both inequalities are positive .

Front 40

One effective technique for solving GMAT inequality problems is to focus on the EXTREME VALUES of a given inequality. This is particularly helpful when solving the following types of inequality problems:

Back 40

(1) Problems with multiple inequalities where the question involves the potential range of values for variables in the problem;
(2) Problems involving both equations and inequalities

Front 41

If 2h + k < 8, g + 3h = 15, and k = 4, what is the possible range of values for g?

Back 41

First, we can simplify the inequality by plugging 4 in for k to simplify the inequality:
2h+4<8 2h<4 h<2
Now, we can use extreme values to combine the inequality with the equation. If we think of h < 2 in terms of extreme values, then h = LT2. We can plug this extreme value into the equation and solve for go
g+3h=15
g +3(LT2) = IS
g+LT6=lS
g=15-LT6
g=GT(9)
g>9
Notice that when we subtract LT6 from IS, we have to CHANGE the extreme value sign from LT to GT. Think of it this way; if we subtract 6 from IS, the result is 9. But if we subtract a number SMALLER than 6 from 15, the result will be LARGER than 9.

Front 42

If x >= 4 + (z + 1)^2, what is the minimum possible value for x?

Back 42

The key to this type of problem-where we need to maximize or minimize when one of the variables has an even exponent-is to recognize that the squared term will be minimized when it is set equal to zero. Therefore, we need to set (z + 1)^2 equal to 0:
(z + 1)^2 = 0
z+1=0
z= -1
The minimum possible value for x occurs when z = -1, and x >= 4 + (-1 + 1)^2= 4 + 0 = 4.
Therefore x ~ 4, so 4 is the minimum possible value for x.

Front 43

Back 43

Front 44

If the length of the side of a cube decreases by two-thirds, by what percentage will the volume of the cube decrease?

Back 44

Front 45

If each number in a sequence is three more than the previous number, and the sixth number is 32, what is the 100th number?

Back 45

Front 46

Back 46

Front 47

8 + LT2

Back 47

Add just like regular numbers:
8 + LT2 = LT10 (i.e., < 10)

Front 48

8 - LT2

Back 48

Subtract and flip the extreme value:
8 - LT2 = GT6 (i.e., > 6)

Front 49

8 * LT2

Back 49

Multiply just like regular numbers:
8 * LT2 = LT16 (i.e., < 16)

Front 50

-7 * LT2

Back 50

Multiply and flip the extreme value:
-7 * LT2 = GT(-14) (i.e., > -14)

Front 51

8 / LT2

Back 51

Divide and flip the extreme value:
8 / LT2 = GT4 (i.e., > 4)
if we know that LT2 is positive

Front 52

LT8 * LT2

Back 52

Multiply just like regular numbers:
LT8 * LT2 = LT16 (i.e., < 16)
if we know that both extreme values are positive