One example using graphical method,
1. Simplify the polynomial to the left side of inequality, and make the right side
zero by eliminating all the terms on the right side using addition or subtraction on both side.
2. Put the polynomial in order from the highest to the lowest and check the highest order has positive coefficient for easy factoring. If the highest order has negative value then multiply by -1 on both sides, and change the direction of inequality.
3. Factor the polynomial and get the values of x's by setting it zero. For example,
For the first order, such as 2x-6, then x=3 which is x-intercept.
For the second order, such as x2- 5x +6, then (x-2)(x-3) where x=2, 3 which are x-intercepts
For the third order, such as x3-2x2-x+3, then (x+1)(x-1)(x-2) where x=-1, 1, 2 which are x-intercepts
........ goes on for the higher order polynomials.
4. Sketch the graph using the x values. For the above example, remember all the highest coefficient should have positive sign for easy factoring,
the first order 2x-6, the line is, from the left to right, going up to the right with x-intercept 3
the second order, x2- 5x +6, the curve is, from the left to right, going down with x-intercept 2, and up with x-intercept 3
the third order, x3-2x2-x+3, the curve is, from the left to right, going up with x-intercept -1, down with x-intercept 1, and up again with x-intercept 2
...... goes on for the higher degrees with same alternate up or down initial patterns. However, there are some graphs having multiple roots, then it touches the x-axis for the even power, crosses the x-axis for odd power of factors.
5. Now taking into consideration of the right side, if the polynomial is greater than 0, then take the x value for top area, if less than zero, then bottom area. If there is equal sign then, add it to the inequality.
I hope this concept works out for you. Do some practice and check it with your graphic calculator. You will catch this idea which can be last forever. Good luck!