Section 3.5, part I: Solving inequalities involving
polynomials and rational
expressions
Okeedokee, then! Now that we know how to handle equalities
involving polynomials and rational expressions ,
it’s just a hop, skip, and half of a jump to solving inequalities involving
these forms.
The procedure we’ll use to solve inequalities involving
polynomials and rational functions is as fol lows :
1. Replace the inequality with the corresponding _________.
2. Solve this _________ using techniques from the previous sections , obtaining
the __________.
(If a rational expression is involved , also compute the values at which the
function is undefined.)
3. These _________ and “holes” in the domain divide the real number line into
various intervals on
which the inequality either holds or doesn’t.
4. On each such interval, choose a “test number” to check whether the inequality
holds on that
interval.
Let’s demonstrate this method with some
Examples. Solve each of the given inequalities
using the method out lined above . (Also, plot a graph
of the expression involved to see if you can see any relationship between the
graph and the solutions to
the inequalities !)
2. -3 < 4x + ≤ 9
(Note: there are two inequalities involved...after solving them one at a time,
the range of values we seek is the _________ of the two separate solutions.)
(Note: in this and other inequalities involving more than one expression, it’s
easiest to move everything to one side of the inequality to begin with.)
7. The force exerted on an object by the Earth’s gravity
is given by F(r) = (GmM)/r2 Newtons, where
G is the gravitational constant, m is the object’s mass (in ki lograms ), M is the
Earth’s mass (in
kilograms), and r is the distance from the object to Earth (in meters).
For what distances r is the force F(r) greater than 10 Newtons? (Note: we can
obtain a solution
in terms of m , M, and G without even knowing what these values are!)
You should now be ready to tackle Homework 3.5.1 on-line;
this homework is due by Friday, October 31st,
but since you will be tested on this material on Thursday, October 30th, I would
strongly recommend
that you attempt the homework before then!
