Equations, Inequalities and VICs

Question 1

Rule for determining whether 2 equations involving 2 variables will be sufficient to solve for the variables..

Answer 1

1. if both of the equations are linear - that is, there are no squared terms (such as x² or y²) and no xy terms - the equations will be sufficient UNLESS the two equations are mathematically identical (x + y = 10 is identical to 2x + 2y=20)
2. If there are ANY non-linear terms in either of the equations (such as x², y², xy or x/y), there will USUALLY be two or more different solutions for each of the variabes and the equations will not be sufficient

Question 2

how to solve absolute value equations

Answer 2

1. isolate expression within the absolute value brackets
2. once we have an equation with |x|=a with a>0, we know that x= +or- a…remove absolute brackets and solve equation for 2 different cases, x=a & x=-a
3. check both solutions by putting back into original equation (always check these)

Question 3

solving problems with exponential expressions on both sides

Answer 3

1. rewrite so same bases are on both sides
2. simplify using rules of exponents
3. eliminate identical bases and rewrite exponents as equation and solve

Question 4

x² - y²

Answer 4

(x+y)(x-y)

Question 5

x² + 2xy +y²

Answer 5

(x + y)(x+y)

(x + y)²

Question 6

x² - 2xy + y²

Answer 6

(x - y)(x - y)

(x - y)²

Question 7

definition of linear sequence

Answer 7

S_n = kn + x

k is constant difference between successive terms and x is some other constant

Question 8

exponential sequence

Answer 8

S_n = x(kⁿ)

x and k are real numbers

each term is equal to the previous term times the constant k

if a ratio of a population in any two consecutive units of time is constant, this implies exponential growth

Question 9

Direct proportionality

Answer 9

two quanitities aways change by the same factor and in the same direction

y/x = k

use ratios to solve

Question 10

indirect proportionality

Answer 10

two quantities change by the reciprocal factors.

y = k/x

use products to solve

Question 11

linear growth or decay

Answer 11

y = mx + b

b is the quanitiy at time = 0

m is the constant rate at which the quantity grows

x will stand for time

Question 12

if x < y

then…

1/x > 1/y when…

1/x > 1/y when…

1/x < 1/y when…

Answer 12

x and y are positive

x and y are negative

when x is negative and y is positive

if we do not know the sign of x or y, we cannot take the reciprocal

in summary, if you know the signs of the variables, you should flip the inequality UNLESS x and y have different signs

Question 13

squaring inequalities (rules)

Answer 13

if both sides are known to be negative, then flip the inequality when you square both sides

if both sides are known to be positive, then do not flip the inequality sign when you square

if one side is positive and one side is negative,then you cannot square

if the signs are unclear, then you cannot square