Vocab: A compound inequality is an inequality made up of two or more inequalities in the same variable.
Examples:
$4x-2>0$ OR $2x+5<-2$
$5x+2>-4$ AND $3x-3<3$
$-x-3<3x-5 \leq- x$
Solutions to Linear Inequalities.
Example: Verify that $x=1$ is a solution to the compound inequality $5x+2>-4$ AND $3x-3<3$.
Example: Verify that $x=2$ is NOT a solution to the compound inequality $5x+2>-4$ AND $3x-3<3$.
Example: Verify that $x=1$ is a solution to the compound inequality $4x-2>0$ OR $2x+5<-2$.
Example: Verify that $x=1$ is a solution to the compound inequality $-x-3<3x-5 \leq -x$.
Finding Solution Sets to Compound Inequalities
Example: Solve the compound inequality $4x-2>0$ OR $2x+5<-2$.
Also, graph the solution set and write it in interval notation.
Solution Sets to Compound Inequalities
Example: Solve the compound inequality $5x+2>-4$ AND $3x-3<3$.
Also, graph the solution set and write it in interval notation.
Solution Sets to Compound Inequalities
Example: Solve the compound inequality $-x-3 < 3x-5 \leq -x$.
Also, graph the solution set and write it in interval notation.
Gremlin Case I: No solution.
Example: Solve the compound inequality $3x-3>0$ AND $4x+1 \leq 5$.
Gremlin Case II: Solution set equals $\mathbb{R}$, or $(-\infty,\infty)$
Example: Solve the compound inequality $3x+5\geq-4$ OR $5x+2\leq -1$.
In Summary
Solutions to OR inequalities:
Solutions to AND inequalities: