Question 384383
Let > imply taller
Let < imply shorter


the first statement says that:


B > j + 3


This means that there is a positive value we can add to j + 3 to make it equal to B.


We'll call that value x.


We get:


B = j + 3 + x


The second statement says that:


P < B + 2


We can replace B with j + 3 + x to make that statement become:


P < j + 3 + x + 2


We can simplify this statement to be equal to:


P < j + x + 5


This means that there is a value we can add to P to make it equal to j + x + 5.


We'll call that value y.


The statement becomes:


P + y = j + x + 5


If y = x + 5, then the statement will become:


P + x + 5 = j + x + 5 


If we subtract (x + 5) from both sides of the equation, we will then get:


P = j


So, if y = x + 5, then we get P = j.


So P = j is possible when y = x + 5.


Going back to the original equation, we have:


P + y = j + x + 5


If we let y be greater than x + 5, then we get another value (call it z), such that:


y = x + 5 + z


Our equation of P + y = j + x + 5 becomes:


P + x + 5 + z = j + x + 5


If we subtract (x + 5) from both sides of this equation, we get:


P + z = j


This implies that P < j


So we can have P = j and we can have P < j


Going back to the original equation again, we have:


P + y = j + x + 5


If we let y be smaller than x + 5, then we get another value (call it w), such that:


y + w = x + 5 which becomes y = x + 5 - w after we subtract w from both sides of the equation.


Our equation becomes:


P + x + 5 - w = j + x + 5


If we subtract x + 5 from both sides of this equation, we get:


P - w = j


If we add w to both sides of this equation, we get:


P = j + w


This implies that P > j


So, we can get all 3 conditions, depending on the relationship between x and y.


They are:


P = j when y = x + 5
P < j when y > x + 5
P > j when y < x + 5


We can put this into numbers in order to confirm that what we have determined algebraically is correct.


Our starting statements are:


B > j + 3
P < B + 2


We translated these into:


B = j + 3 + x (x > 0)
P + y = B + 2 (y > 0)


Let's let x = 5 and y = 10 (y = x + 5)


We get:


B = j + 3 + 5 which becomes B = j + 8
P + 10 = B + 2 which becomes P = B - 8


Substitute j + 8 for B and we get P = j + 8 - 8 which becomes P = j.


When y = x + 5, we get P = j.


Back to the original equations:


B > j + 3
P < B + 2


We translated these into:


B = j + 3 + x (x > 0)
P + y = B + 2 (y > 0)


Let's let x = 5 and y = 11 (y > x + 5)


We get:


B = j + 3 + 5 which becomes B = j + 8
P + 11 = B + 2


Substitute j + 8 for B in the second equation and we get:


P + 11 = j + 8 + 2 which becomes:
P + 11 = j + 10
Subtract 10 from both sides of the equation to get:
P + 1 = j
This implies that P is less than j (P < j).


Back to the original equations:


B > j + 3
P < B + 2


We translated these into:


B = j + 3 + x (x > 0)
P + y = B + 2 (y > 0)


Let's let x = 5 and y = 9 (y < x + 5)


We get:


B = j + 3 + 5 which becomes B = j + 8
P + 9 = B + 2


Substitute for B in the second equation to get:


P + 9 = j + 8 + 2 which becomes:
P + 9 = j + 10


Subtract 9 from both sides of the equation to get:


P = j + 1


This implies that P > j


Bottom line:


P can be taller than or equal to or shorter than j.


This all depends on the relationship between x and y.


In the translation of the equation B > j + 3 to B = j + 3 + x, x is the amount that B is greater than (taller than) j + 3.


In the translation of the equation P < B + 2, to P + y = B + 2, y is the amount that P is smaller than (shorter than) B + 2.


Making the inequality an equality allows the problem to be solved.


The creation of the difference variables of x, y, z, w allowed the equality to be modeled from the inequality.


x,y,z,w are all assumed to be greater than 0.