Summary on revision on Linear Inequalities in One Variable
- The open interval $(a, b)$ with endpoints ( a ) and ( b ) represents the set of all real numbers ( x ) where $( a < x < b )$.
- The closed interval $[a, b]$ with endpoints (a) and (b) is the set of all real numbers (x) where $(a \leq x \leq b)$.
- The half-open or half-closed interval $(a, b]$ with endpoints ( a ) and ( b ) is the set of all real numbers ( x ) where $( a < x \leq b )$.
- The half-open or half-closed interval $[a, b)$ with endpoints (a) and (b) includes all real numbers (x) where $(a \leq x < b)$.
- Some Properties of Inequalities
- If $( a > b )$, then $( a + c > b + c )$
- If $( a > b )$, then $( a – c > b – c )$
- If $( a > b )$ and $( m > 0 )$, then $( ma > mb )$ and $( \frac{a}{m} > \frac{b}{m} )$
- If $( a > b )$ and $( m < 0 )$, then $( ma < mb )$ and $(\frac{a}{m} < \frac{b}{m} )$
- For any positive real number aaa, the set of solutions for:
- i. the inequality $\left| x \right| < a$ is $-a < x < a$ and
- ii. the inequality $\left| x \right| > a$ is $x < – a$ or $x > a$.