Question 1153658
The difference between a polynomial or rational equation and polynomial or rational inequality:

An equation has the EQUALITY sign  " = "  between two parts (between two expressions).

A polynomial function is a function of the form:

{{{a[n]x^n}}}+....+{{{a[1]*x+a[0] }}}where  {{{n}}} is a non-negative integer and {{{a[n]<>0}}}
A polynomial of degree {{{n }}}has at most {{{n}}} real zeros and {{{n-1}}} turning points.


The difference between a polynomial or rational equation and polynomial or rational inequality:

A polynomial function is a function of the form:

{{{a[n]x^n}}}+....+{{{a[1]*x+a[0]}}} where  {{{n}}} is a non-negative integer and {{{a[n]<>0}}}

A polynomial of degree {{{n}}} has at most {{{n}}} real zeros and {{{n-1}}} turning points.

First, the end behavior of a polynomial is determined by its {{{degree}}} and the {{{sign }}}of the {{{lead}}}{{{ coefficient}}}. 

The degree of a polynomial function determines the end behavior of its graph. If the degree of a polynomial is even, then the end behavior is the same in both directions. If the degree of a polynomial is odd, then the end behavior on the left is the opposite of the behavior on the right.

A rational equation is just a quotient of two polynomials. Quotient here just means “fraction.” So, if {{{r(x)}}} is a rational expression, then it can be written as:

{{{r(x)=p(x)/q(x)}}} where {{{p(x) }}}and {{{q(x)}}} are both polynomials.
 
A rational function may have a {{{vertical}}} asymptote whenever {{{q(x)=0}}} which {{{restricting}}} the {{{domain}}} of the function.

The graphs of rational functions may have vertical asymptotes only where the denominator is zero, and  a {{{horizontal}}} asymptote (a horizontal line that the graph approaches as the input increases or decreases without bound).

The end behavior of the graph of a rational function is determined by the {{{degrees}}} of the {{{polynomials}}} in the {{{numerator}}} and {{{denominator}}}.

The graphs of rational functions may have vertical asymptotes only where the denominator is zero. 


Two differences  difference between a {{{polynomial}}} or{{{ rational}}} {{{inequality }}}: 
First, an inequality has the INEQUALITY sign   " < ",  or  " <= ",  or " > ",  or  " >= "   between two parts (between two expressions).

A polynomial inequality (PI) can {{{always}}} be replaced by a PI with one side zero. That is,{{{ p(x)>q(x)}}} is the same as {{{p(x)-q(x)>0}}}. 

 A rational inequality (RI) {{{can}}}{{{ not}}}, because you {{{do}}}{{{ not}}} {{{know}}} the {{{sign}}} of the {{{denominator}}}; 

{{{a(x)/b(x)>c(x)/d(x)}}} can not safely be replaced by 

{{{(a(x)d(x)-b(x)c(x))/(b(x)d(x))>0}}} 

because {{{b(x)}}} and/or {{{d(x)}}} might change signs unexpectedly.