Download To solve Rational Inequalities:

To graph Rational Functions:
(1) Find all holes that may appear in your graph by finding the common factor(s) that
appears in the N(x) and D(x). Set the common factors = zero and solve for your x
coordinate of the hole. Plug in the x values into the simplified function to solve for y.
Go to your graph and put an open circle on the graph at those x-values
(2) Find all the x-intercepts by finding all the real zeros of N(x) that are not also zeros
of D(x): Set numerator equal to 0 and solve for x. Graph the value(s) on the x-axis.
(3) Find the y-intercept by evaluating R(0) (Plug in 0 for x and solve for y.) Graph the
value on the y-axis.
(4) Find all the vertical asymptotes by setting denominator equal to 0 and solve for x.
Graph the value(s) found as vertical dashed lines
(5) Find any horizontal asymptote by comparing the degree of N(x) with the degree
of D(x).If the degree of N(x) < D(x), then y = 0 is your horizontal asymptote. If the
degree of N(x) = D(x), then y = the ratio of the lead coefficients is your horizontal
asymptote. If the degree of N(x) > D(x), then there is no horizontal asymptote. Graph
the value found as a horizontal dashed line
(6) Find any slant/oblique asymptote. A slant asymptote will occur if the degree of
N(x) is exactly one degree more than the degree of D(x).How? Use long division.
Your slant/oblique asymptote will be y = the quotient, without the remainder. Graph
the equation found as a diagonal dashed line.
(7) Make a table of values by choosing x values on either side of your vertical
asymptotes. Plug them into your rational equation and evaluate for y. Plot all these
points found on your graph.
(8) Complete your graph. You should connect your points and use the asymptotes as
magnets but keep in mind the following: Your graph should never cross a vertical
asymptote and it should approach your horizontal or slant/oblique asymptote at both
ends.
Example:
To find inverses:
(1) Change f(x) to y.
(2) Switch all x and y variables
(3) Solve for y.
(4) Change y to f-1(x)
Example:
To simplify rationals (addition and subtraction):
(1) Find lowest common denominator EVEN FOR NON-FRACTIONS
(2) Do NOT multiply each term by lowest common denominator, but multiply each
term (top and bottom) to make the term have the common denominator
(3) Add or subtract the numerator and keep the same common denominator
Example:
To simplify rationals (multiplication/division):
(1) Factor numerators and denominators
(2) If dividing fractions: keep, change, flip
(3) Cancel out any factors appearing in numerator and denominator
Example:
To solve rationals:
(1) Factor numerators and denominators.
(2) Multiply ALL terms by lowest common denominator
(3) Cancel out common factors. This should leave you with no denominators
(4) Combine like terms and solve for x.
Example:
To solve Rational Inequalities:
(1) Determine any values that make the denominator equal 0.
(2) Write the inequality as an equation, and solve the equation
(3) On a number line, mark each of the critical values from steps 1 and 2.
These values will create intervals on the number line.
(4) Select a test point in each interval, and check to see if that test point
satisfies the inequality. (Find the intervals which satisfy the inequality).
(5) Mark the number line to reflect the values and intervals that satisfy
the inequality.
(6) State your answer using the desired form of notation.
Example:
To decompose partial fractions:
(1) Factor denominator.
(2) Set original fraction = A over the first factor + B over the second factor…
(3) Multiply all terms by LCD.
(3) Determine what values of X will cancel A, B, etc.
(4) Plug in the X values from step 4, separately, into the equation to determine values
for A, B, etc.
(5) Put the values for A, B, etc. over their factors from step 2.
Example: