Property 3
In solving an inequality system always positive expressions are cancelled.
- \(|f(x)|\ge0\) — Expression of absolute value
- \(n\in\mathbb{N},\ [f(x)]^{2n}\ge0\) — Squared expression
- \(a^x>0,\ a>0\) — Exponential function
- \(f(x)=ax^2+bx+c>0,\ \Delta<0,\ a>0\)
Quadratic equation without root. If “a” is a negative number it will be cancelled and only the direction of the sign will change.
\(n\in\mathbb{N}\)
\(f(x)>0\Rightarrow[f(x)]^{2n-1}>0\)
\(f(x)<0\Rightarrow[f(x)]^{2n-1}<0\)
As the signs of f(x) and [f(x)]2n−1 are the same the power of the exponential expression will be cancelled.
If expressions can be simplified then they are simplified but only the roots will be analysed.
Note: The roots of the cancelled expressions should be analyzed.
Example
\[ \samefrac{|x-3|\cdot3^x\cdot(x-2)\cdot(x+7)^4\cdot(x-9)} {(x^2+x+7)\cdot(x+5)^{99}\cdot(x-9)}\ge0 \]Answer
Always-positive factors are cancelled. Common numerator and denominator factors are simplified, but their roots are still analysed.
\(\samefrac{x-2}{x+5}\ge0\)
| x | −∞ | −5 | 2 | +∞ | |
|---|---|---|---|---|---|
| f(x) | + | ○ | − | ○ | + |
\((−\infty,-5)\cup[2,\infty)\)
\(x=3\qquad x=-7\qquad x\ne9\)
\(\Rightarrow ((-\infty,-5)\cup[2,\infty))\setminus\{9\}\)
1.
\(\samefrac{(x-2)(x+4)^2}{x^2}<0\)
\(\Rightarrow S.S.=?\)
2.
\(\samefrac{(x^2-4x+4)x^3}{3-x}\ge0\)
\(\Rightarrow S.S.=?\)
3.
\(\samefrac{x^2-6x+5}{(x-4)^2}<0\)
\(\Rightarrow S.S.=?\)