System with three conditions

With three conditions the procedure does not change: you solve the individual inequalities, mark them on the real line and look for the single stretch common to all of them.

Example — System with three conditions

Solve $\begin{cases} x^2-x-6 \ge 0 \\ x+1 > 0 \\ x-5 < 0 \end{cases}$

D$_1$: $(x-3)(x+2)\ge 0 \iff x\le -2 \;\text{oppure}\; x\ge 3$.

D$_2$: $x>-1$. D$_3$: $x<5$.

The only stretch common to the three conditions is the interval $[3,5)$.

Solution: the only common stretch is $\boxed{3\le x < 5}$.

Links

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