We can solve inequalities just by looking at the graph of a function.
Example 1: Solve the following inequality x3 + 2x2 - 3x - 6 < 0.
The main connection that needs to be made here is that this question is the same as:
Find the intervals on the x axis over which the polynomial funtion
f(x) = x3 + 2x2 - 3x - 6 is negative.
(see Negative Functions)
- Graph the function
-
The intervals on the x-axis where the function is negative are highlighted
in red. These intervals provide the solutions to the inequality.
- The red interval on the left goes from negative infinity to the first
negative root (see Zoom on right). This interval is (-inf, -2). The second red
interval goes from the next negative root to the positive root. This interval
is (-1.73205080757, 1.73205080757) (see
Finding Roots of a Function).
Example 2: Solve the following inequality
> 0
This question is the same as:
Find the intervals on the x axis over which the function
f(x) = is positive
- Graph the function;
- The parts of the graph above highlighted in blue are the parts of the graph
that are positive.
The intervals on the x-axis where the function is positive are highlighted
in red. These are the intervals that satisify the inequality.
- The intervals are (-inf, -3/2) and (3/2, inf). (see
Finding Roots of a Function).
Example 3: Solve the following inequality x3 + 2x2 < 3x + 6 .
By subtracting the right hand side from both sides of the inequality we see that this question is the same as Example 1:
Solve the inequality x3 + 2x2 - 3x - 6 < 0.
You could even avoid the algebra by entering the left hand side as the function y1 and the right hand side as y2. Unselect these and enter for y3 the difference y1 - y2. This graph will be the same as the one in Example 1.