System of two inequalities

When two or more inequalities must be satisfied simultaneously, we speak of a system of inequalities. The solution is the intersection of the solutions of the individual inequalities: they are represented on the real line and the common stretches are taken.

Example — System of two inequalities

Solve the system $\begin{cases} x^2-4 > 0 \\ x-1 < 0 \end{cases}$

D$_1$: $x^2-4>0 \iff (x-2)(x+2)>0 \iff x<-2 \;\text{or}\; x>2$.

D$_2$: $x<1$.

Graphical representation of the solutions:

The solution of the system (S) is the intersection of the stretches of D$_1$ and D$_2$.

Solution: $\boxed{x<-2}$.

Links

Topics: Inequalities
Concepts: Interval · System of inequalities
Skills: Interpreting a graph · Solving inequalities · Solving systems