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Polynomial and Rational Inequalities
This section will explore how to solve inequalities that are either in rational or polynomial form.
Example 1. Solve the equation x^2 < x + 6 and graph the solution on a number line.
Step 1. Write the equation in standard form. 2 2 2
x x x x x x x x
Step 2. Find the zeros of the equation (the values of x that make the equation equal to zero).
2 2
x x x x x x x or x x or x
Step 3. Organize data.
The values found in Step 2, (3, 0) and (-2, 0) are plotted on a number line and the number line is divided into sections (intervals) with boundaries passing through these points:
These values are known as critical points because the boundaries of any intervals pass through them.
Step 4. Test each interval for its sign values.
To find the sign value of each interval, select any point within the interval (EXCEPT THE CRITICAL POINTS) and substitute the value for x in the factored form of the polynomial.
Test point for the interval (-∞, -2) : x = -
x − x + − − − + − − =
The product found was positive, therefore the interval is also positive. Test point for the interval (-2, 3) : x = 0
x − x + − + − = − The product found was negative, therefore the interval is also negative. Test point for the interval (3, ∞) : x = 5
x − x + − + = The product found was positive, therefore the interval is also positive.
Step 2. Find the critical points.
Note that in addition to the values that make the equation equal to zero, in rational expressions a critical point will involve those values that cannot be used (i.e. the values that would make the denominator zero).
(^8 ) 5 8 0 5 0 8 5
x x x or x x or x
∴ The critical points for this equation are (8, 0) and (5, 0)
Step 3. Organize data.
Since the equation will allow zero to be a solution, the critical point (8, 0) is part of the solution set, whereas the point (5, 0) is not. The reason is the substitution of x = 5 would result in a division by zero. As can be seen on the number line (-∞, 5) , (5, 8] and [8, ∞) are the intervals to be inspected.
Step 4. Test the intervals for the sign values.
Test point for the interval (-∞, 5) : x = 0
x x
This value is negative so the intervals values are also negative. Therefore this interval is part of the solution. Test point for the interval (5, 8] : x = 6
x x
This value is positive so the intervals values are also positive. Therefore this interval is not part of the solution.
Test point for the interval [8, ∞) : x = 10
x x
This value is negative so the intervals values are also negative. Therefore this interval is part of the solution.
