# Solving systems of quadratic inequalities. How to solve a system of quadratic inequalities, using the double-number-line?

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There are 2 main methods to solve a system of 2 quadratic inequalities:

1. The algebraic method by using a sign chart

2. The innovative method using the double-number line.

There are 2 main methods to solve a system of 2 quadratic inequalities:

1. The algebraic method by using a sign chart

2. The innovative method using the double-number line.

##### 1 Answer

We can use the double-number-line to solve any system of 2 or 3 quadratic inequalities in one variable (authored by Nghi H Nguyen)

#### Explanation:

Solving a system of 2 quadratic inequalities in one variable by using a double number-line.

**Example 1.** Solve the system:

First solve f(x) = 0 --> 2 real roots: 1 and -3

Between the 2 real roots, f(x) < 0

Solve g(x) = 0 --> 2 real roots: -1 and 5

Between the 2 real roots, g(x) < 0

Graph the 2 solutions set on a double number-line:

f(x) -----------------------------0 ------ 1 ++++++++++3 --------------------------

g(x) ------------------ -1 ++++0+++++++++++++++ 3 ++++++++5 ----------

By superimposing, we see that the combined solution set is the open interval (1, 3).

**Example 2** . Solve the system:

Solve f(x) = 0 --> 2 real roots: -1 and 5

Between the 2 real roots, f(x) < 0

Solve g(x) = 0 --> 2 real roots: 1 and 2

Out side the 2 real roots, g(x) > 0

f(x) --------------------- -1 ++++ 0 ++++++++++++++++++++ 5 ---------------

g(x) ++++++++++++++++++++++++ 1 -------2 +++++++++++++++++++++

By superimposing, we see that the combined solution set are the

open intervals: (- 1, 1) and (2, 5)