Equazione fratta e disequazione letterale intera
Equazione fratta:
[math]\frac{x-1}{2a+1}+\frac{x-x^{2}-2}{4a^{2}-1}=\frac{x(1-x)}{(2a+1)(2a-1)} \\
\frac{x-1}{2a+1}+\frac{-x^{2}+x-2}{(2a+1)(2a-1)}=\frac{x(1-x)}{(2a+1)(2a-1)} \\
C.E. : a \not{=} +\frac{1}{2} , a \not{=} -\frac{1}{2} \\
m.c.m. (2a+1)(2a-1) \\
(x-1)(2a-1)-x^{2}+x-2=x(1-x) \\
2ax-x-2a+1-x^{2}+x-2=x-x^{2} \\
2ax-x=2a+1 \\
x(2a-1)=2a+1[/math]
\frac{x-1}{2a+1}+\frac{-x^{2}+x-2}{(2a+1)(2a-1)}=\frac{x(1-x)}{(2a+1)(2a-1)} \\
C.E. : a \not{=} +\frac{1}{2} , a \not{=} -\frac{1}{2} \\
m.c.m. (2a+1)(2a-1) \\
(x-1)(2a-1)-x^{2}+x-2=x(1-x) \\
2ax-x-2a+1-x^{2}+x-2=x-x^{2} \\
2ax-x=2a+1 \\
x(2a-1)=2a+1[/math]
Discussione:
->
[math]1. a=\frac{1}{2} ; 0x=2 \to Impossibile \\
2. a \not{=} \frac{1}{2} ; \frac{x(2a-1)}{(2a-1)}=\frac{2a+1}{(2a-1)} ; x=\frac{2a+1}{2a-1}[/math]
.2. a \not{=} \frac{1}{2} ; \frac{x(2a-1)}{(2a-1)}=\frac{2a+1}{(2a-1)} ; x=\frac{2a+1}{2a-1}[/math]
Disequazioni letterali intere:
-
[math](x+a)(x-a)+1-x^{2}>2(1-a^{2})-ax \\
x^{2}-a^{2} +1-x^{2}>2-2a^{2}-ax \\
ax>1-a^{2} \\
ax>(1+a)(1-a)[/math]
x^{2}-a^{2} +1-x^{2}>2-2a^{2}-ax \\
ax>1-a^{2} \\
ax>(1+a)(1-a)[/math]
Discussione:
->
[math]1. a=0 ; 0x>1 ; \not{∃}x \in \mathbb{R} \\
2. a>0 ; x>\frac{(1+a)(1-a)}{a} ; x>\frac{1-a^{2}}{a} \\
3. a
2. a>0 ; x>\frac{(1+a)(1-a)}{a} ; x>\frac{1-a^{2}}{a} \\
3. a
-
[math]a (x-3)
ax - 3a
ax - x
x (a - 1)
Discussione:
->
[math]1. a=1 ; 0x
2. a>1 ; x
3. a\frac{3a-1}{a-1}[/math]
-
[math](a-1)x-(1-a)(1+a)
(1-a)x>-(1-a)(1+a)[/math]
Discussione:
->
[math]1. a=1 ; 0x>0 ; \not{∃} x \in \mathbb{R} \\
2. a>1 ; x 3. a-(1+a)[/math]
2. a>1 ; x 3. a-(1+a)[/math]
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