Interactive Tutorials
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Example 1 : Solve graphically and analytically the quadratic inequality
 x ^{2} + 3x + 4 < 0
Solution to Example 1:
Graphical solution:
Use the applet to set coefficients a = 1, b = 3 and c = 4 and graph the equation y =  x ^{2} + 3x + 4. The solution set to the inequality  x ^{2} + 3x + 4 < 0 correspond to the x coordinates of the points on the graph for which y < 0 BLUE. We have two intervals for which y < 0:
(∞ , 1) U (4 , +∞)
Analytical solution:

Factor the left hand term of the given inequality
 x ^{2} + 3x + 4 = (x + 1)(x + 4)

To solve the given inequality, we study the sign of
(x + 1)(x + 4)
on the intervals (∞ , 1), (1 , 4) and (4 , +∞) where 1 and 4 are the zeros of (x + 1)(x + 4).
let x = 2 a value of x on the interval (∞ , 1). For this value  x ^{2} + 3x + 4 = (2)^{ 2} + 3(2) + 4 = 6 is negative. The interval (∞ , 1) is one solution set. Let x = 0 on the interval (1 , 4). For this value  x ^{2} + 3x + 4 = 4 is positive. x = 5 is in the interval (4 , +∞). For this value of x,  x ^{2} + 3x + 4 = 6 is negative. The interval (4 , +∞) is a solution set for the given inequality.
Hence the solution set of the inequality is given by the union of all intervals for which  x ^{2} + 3x + 4 is negative:
(∞ , 1)U (4 , +∞)

Both the graphical and analytical methods give the same answer.
Example 2 : Solve graphically and analytically the equation
x ^{2} + 4x  5 > 0
Solution to Example 2:
Graphical solution:
Use the applet to set coefficients a = 1, b = 4 and c = 5 and graph the equation y = x ^{2} + 4x  5. This inequality has no solutions since the whole graph is below the x axis and therefore y < 0 everywhere.
Analytical solution:

x ^{2} + 4x  5 cannot be factored over the real numbers. Therefore x ^{2} + 4x  5 has no zeros and its sign does not change. To find the sign of x ^{2} + 4x  5 you need to evaluate it for one single value of x. Let us evaluate x ^{2} + 4x  5 at x = 0
(0) ^{2} + 4(0)  5 =  5.

x ^{2} + 4x  5 is always negative hence the inequality x ^{2} + 4x  5 > 0 has no solutions.
Exercises: Solve graphically (using the applet) and analytically the following quadratic inequalities.
1: x^{ 2}  4 x < 5
2: x^{ 2}  2 x + 8 >= 0
3: x^{ 2}  3 x <= 0
Solutions to Above Exercises:
1: (∞ , 5) U (1 , +∞)
2: (∞ , +∞)
3: [0 , 3]
More references and links on how to Solve Equations, Systems of Equations and Inequalities.