# Quadratic Inequalities - Examples, Exercises and Solutions

The quadratic inequality shows us in which interval the function is positive and in which it is negative - according to the inequality symbol. To solve quadratic inequalities correctly, it is convenient to remember two things:

1. Set of positivity and negativity of the function:
Set of positivity - represents the $X$s in which the graph of the parabola is above the $X$ axis, with $Y$ value positive.
Set of negativity - represents the $X$s in which the graph of the parabola is below the $X$ axis, with $Y$ value negative.
2. Dividing by a negative term - reverses the sign of the inequality.

#### Method to solve the quadratic inequality:

1. We will carry out the transposition of members and isolate the quadratic equation until one side equals 0. Remember that when we divide by a negative term, the inequality is reversed.
2. Let's draw a diagram of the parabola - placing points of intersection with the $X$ axis and identifying the maximum and minimum of the parabola.
3. Let's calculate the corresponding interval according to the exercise and the diagram.
Quadratic equation $>0∶$ Set of positivity
Quadratic equation $<0∶$ Set of negativity

### Suggested Topics to Practice in Advance

1. Solution of a system of equations - one of them is linear and the other quadratic

## Examples with solutions for Quadratic Inequalities

### Exercise #1

Solve the following equation:

x^2+4>0

### Video Solution

All values of $x$

### Exercise #2

Solve the following equation:

-x^2+2x>0

0 < x < 2

### Exercise #3

Solve the following equation:

-x^2-9>0

### Video Solution

There is no solution.

### Exercise #4

Solve the following equation:

x^2+9>0

### Video Solution

All values of $x$

### Exercise #5

Solve the following equation:

x^2-9<0

-3 < x < 3

### Exercise #6

Solve the following equation:

-x^2+3x+4>0

-1 < x < 4

### Exercise #7

Solve the following equation:

x^2-3x+4<0

### Video Solution

There is no solution.

### Exercise #8

Solve the following equation:

x^2-6x+8<0

2 < x < 4

### Exercise #9

Solve the following equation:

x^2+4>0

### Video Solution

All values of $x$

### Exercise #10

Solve the following equation:

x^2-8x+12>0

x < 2,6 < x

### Exercise #11

Solve the following equation:

x^2+4x>0

x < -4,0 < x

### Exercise #12

Solve the following equation:

x^2+4x>0

x < -4,0 < x

### Exercise #13

Solve the following equation:

x^2-2x-8>0

### Exercise #14

Solve the following equation:

x^2-25<0

-5 < x < 5

### Exercise #15

Solve the following equation:

-x^2-25<0

### Video Solution

All values of $x$