Solving systems of quadratic inequalities. How to solve a system of quadratic inequalities, 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.

Oct 9, 2017

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:
$f \left(x\right) = {x}^{2} + 2 x - 3 < 0$ (1)
$g \left(x\right) = {x}^{2} - 4 x - 5 < 0$ (2)
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:
$f \left(x\right) = {x}^{2} - 4 x - 5 < 0$
$g \left(x\right) = {x}^{2} - 3 x + 2 > 0$
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)