Property 1
Basic Inequality
\(A(x)<B(x)\)
- \(A(x)+C<B(x)+C\)
\(A(x)-C<B(x)-C\) - \(A(x)\cdot C<B(x)\cdot C\qquad C\in\mathbb{R}^+\)
\(\samefrac{A(x)}C<\samefrac{B(x)}C\) - \(A(x)\cdot C>B(x)\cdot C\qquad C\in\mathbb{R}^-\)
\(\samefrac{A(x)}C>\samefrac{A(x)}C\)
The same number can be added to or subtracted from both sides of an inequality. Both sides can be multiplied or divided by the same positive number direction of inequality stays the same. If it is multiplied or divided by a negative number the direction of the inequality is reversed.
\((-\infty,a)\)
\((-\infty,a]\)
\((a,+\infty)\)
\([a,+\infty)\)
\((a,b)\)
\([a,b]\)
\((a,b]\)
\([a,b)\)
\((-\infty,+\infty)=\mathbb{R}\)
Let \(a<b\)
\((-\infty,a)\cup(b,\infty)=\mathbb{R}\setminus[a,b]\)
\((-\infty,a]\cup[b,\infty)=\mathbb{R}\setminus(a,b)\)
\((-\infty,a)\cup[b,\infty)=\mathbb{R}\setminus[a,b)\)
\((-\infty,a]\cup(b,\infty)=\mathbb{R}\setminus(a,b]\)
1.
\(x\in\mathbb{Z}\)
\(2x-1<5\)
\(\Rightarrow\max(x)=?\)
2.
\(x\in\mathbb{Z}\)
\(3x-3>7\)
\(\Rightarrow\min(x)=?\)
3.
\(x\in\mathbb{Z}\)
\(2(x-1)+x-3\ge x+1\)
\(\Rightarrow\min(x)=?\)