Section 3.5
Polynomial and Rational
Inequalities
Inequalities
Polynomial Inequality
ex.
3x  4x  7  0
3
Rational Inequality
ex.
2x  1
0
7x  10
Consider P(x) = x³ + x² - 6x
• Find the zeros:
Zeros:
All are multiplicity
Graph will
Critical Values
The values of 0, -3, and 2 (the zeros) are
considered to be critical values.
Critical values are values of x for which
P(x) = 0 is undefined or equal to zero.
Graph P(x) = x³ + x² - 6x on a graphing
calculator and solve the following.
1.) P(x) = 0
2.) P(x) > 0
3.) P(x) < 0
4.) P(x) > 0
5.) P(x) < 0
P(x) = x³ + x² - 6x
Polynomial Inequalities
• A quadratic inequality can be written in the
form
ax2 + bc + c > 0, where the symbol could be
replaced with either <, , or .
• A quadratic inequality is one type of
polynomial inequality.
• Examples:
2 x  x  2  4
4
2
3
x60
4
8x  2 x  6 x  5
3
2
Steps for Solving Polynomial Inequalities
1. Rewrite the inequality so that there is a zero on the
right side.
2. Find the x-intercepts (zeros), if any. (Solve the
polynomial equation.) The zero(s) are called critical
values.
3. The x-intercepts divide the x-axis into intervals.
Select test values in each interval and determine the
sign of the polynomial on that interval.
4. The solution will be those intervals in which the
function has the correct signs satisfying the
inequality.
Example
Solve: 4x3  7x2  15x.
We need to find all the zeros of the function
so we solve the related equation.
The zeros are
Thus the x-intercepts of the graph are
Example continued
• The zeros divide the x-axis into four intervals. For all xvalues within a given interval, the sign of 4x3  7x2  15x 
0 must be either positive or negative. To determine which,
we choose a test value for x from each interval and find
f(x).
• Since we are solving 4x3  7x2  15x  0, the solution set
consists of only two of the four intervals, those in which
the sign of f(x) is negative.
Steps for Solving Rational Inequalities
1. Find an equivalent inequality with 0 on one side.
2. Change the inequality symbol to an equals sign and
solve the related equation. (Zeros)
3. Find the values of the variable for which the related
rational function is not defined.
4. The numbers found in Steps 2 & 3 are called critical
value. Use the critical values to divide the x-axis into
intervals, then test an x-value from each interval to
determine the function’s sign in that interval.
5. Select the intervals for which the inequality is satisfied
and write interval notation for the solution set. If the
inequality symbol is < or > , then the zeros (Step 2)
should be included in the solution set. The x-values
for which the function is not defined (Step 3) are never
included in the solution set.
Example
• Solve
x3
0
2
x 1
• The denominator tells us that f(x) is not
defined for x = and x= .
(These are critical values.)
Example continued
• Next, solve f(x) = 0.
Example continued
• The critical values are 3, 1, and 1.
• These values divide the x-axis into four intervals.
• We use a test value to determine the sign of f(x) in each
interval.