27-Resolvendo Inequa\303\247\303\265es N\303\243o- Lineares
Resolvendo todos os tipos de inequa\303\247\303\265es de valor absoluto b\303\241sico.
<Text-field style="Heading 1" layout="Heading 1"><Font size="14" encoding="UTF-8">0-C\303\263digo</Font></Text-field>restart: with(plots):PolyInequal := proc(expr)
local k,j,s,t,a,b,minS,maxS,L;
for k from 1 to nops(expr) do
s||k := solve( op(k, expr) = 0); od:
minS := s||1: maxS := s||1:
for k from 1 to nops(expr) do
if( s||k < minS) then minS := s||k fi:
if( s||k > maxS) then minS := s||k fi:
od:
L := (maxS - minS): a:= floor( minS - L/4): b:= ceil( maxS + L/4):
for k from 1 to nops(expr) do
printf("\134n ");
for j from a to s||k do printf("|----") od;
for j from s||k to b do printf("|++++"); od;
printf("|\134t\134t\134t\134t%A\134n",(op(k,expr)));
od;
for j from a-1 to b+1 do
if( j < 0 ) then printf("\134t%d ", j); else printf("\134t %d ", j); fi: od;
for k from 1 to 1 do
printf("\134n ");
for j from a-1 to b do
subs(x = j +.5, expr ): t := round(%/abs(%)):
if ( t > 0 ) then printf("|++++"); else printf("|----") fi; od:
printf("| %A", expr);
od;
print(` `); print(solve( expr > 0)); print(` `);
end proc:
QuoInequal := proc(top, bottom )
local k,j,s,t,a,b,minS,maxS,L,expr;
expr := top*bottom;
for k from 1 to nops(expr) do
s||k := solve( op(k, expr) = 0); od:
minS := s||1: maxS := s||1:
for k from 1 to nops(expr) do
if( s||k < minS) then minS := s||k fi:
if( s||k > maxS) then minS := s||k fi:
od:
L := (maxS - minS): a:= floor( minS - L/4): b:= ceil( maxS + L/4):
for k from 1 to nops(expr) do
printf("\134n ");
for j from a to s||k do printf("|----") od;
for j from s||k to b do printf("|++++"); od;
printf("|\134t\134t\134t\134t%A\134n",(op(k,expr)));
od;
for j from a-1 to b+1 do
if( j < 0 ) then printf("\134t%d ", j); else printf("\134t %d ", j); fi: od;
for k from 1 to 1 do
printf("\134n ");
for j from a-1 to b do
subs(x = j +.5, expr ): t := round(%/abs(%)):
if ( t > 0 ) then printf("|++++"); else printf("|----") fi; od:
printf("| %A", expr);
od;
end proc:<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8"> 1. Inequa\303\247\303\265es Lineares </Font><Font size="14" style="Text" bold="true"> </Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
Pode ser uma boa id\303\251ia o conceito b\303\241sico de resolu\303\247\303\243o de uma equa\303\247\303\243o linear simples.(x-3) > 0; solve(%,x);(x+7) > 0; solve(%,x);(x-11) >= 0; solve(%,x);(x-4) <= 0; solve(%,x);
\303\211 um pouco mais complicado quando n\303\243o estamos apenas lidando com |x| e sim com o valor absoluto de uma express\303\243o linear de x.abs(x - 3) = 10; x = {solve(%,x)};<Text-field style="Heading 2" layout="Heading 2"> 2. Produto do Termos <Font size="14" style="Text" bold="true" foreground="[0,128,128]"> </Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
Uma inequa\303\247\303\243o polinomial com dois ou mais fatores requer um pouco mais de trabalho na sua resolu\303\247\303\243o. (x-4)*(x+3) > 0;
Podemos ter uma id\303\250ia melhor do que est\303\241 acontecendo observando alguns valores.for k from -7 to 9 do
`x = `,k,` Valor = `,(k-4)*(k+3); od;Podemos ver que os sinais s\303\243o positivos, depois negativos e depois positivos novamente. H\303\241 uma intera\303\247\303\243o entre os sinais dos dois termos.
A maneira de resolver isso \303\251 criar um gr\303\241fico de sinais.
1. Fatore a express\303\243o (se ainda n\303\243o estiver fatorada)
2. Fa\303\247a um gr\303\241fico dos valores dos sinais para cada fator acima de uma linha num\303\251rica comum
3. Multiplique os sinais para obter um gr\303\241fico de sinais
4. "Leia" o gr\303\241fico de sinais - veja a inequa\303\247\303\243o original e verifique se \303\251 >, >=, < ou <= 0, l
e ent\303\243o procure pela parte apropriada do gr\303\241fico
Vamos criar um gr\303\241fico de sinais para o problema acima.
PolyInequal( (x-4)*(x+3));
Vejamos alguns outros exemplos. Mesmo que tenha mais fatores, podemos usar o mesmo m\303\251todo. a \303\272nica diferen\303\247a \303\251 que teremos linhas adicionais - de maneira que tenhamos um gr\303\241fico de sinais para cada fator. Esses exemplos t\303\252m tr\303\252s fatores.(x-3)*(x+1)*(x+5) > 0;
PolyInequal( (x-3)*(x+1)*(x+5)); (x-7)*(x+1)*(x+2) > 0;
PolyInequal( (x-7)*(x+1)*(x+2) );
Estes exemplos t\303\252m tr\303\252s ou mais fatores.
(x-3)*(x+7)*(x+1)*(x+2) > 0;
PolyInequal( (x-3)*(x+4)*(x+1)*(x+2) );(x-4)*(x-3)*(x-1)*(x+1)*(x+2)*(x) > 0;
PolyInequal( (x-4)*(x-3)*(x-1)*(x+1)*(x+2)*(x) );<Text-field style="Heading 2" layout="Heading 2"> 3. Quociente dos Termos<Font size="14" style="Text" bold="true" foreground="[0,128,128]"> </Font><Font size="14" style="Text" bold="true"> </Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
Agora vamos ver as inequa\303\247\303\265es com quocientes.(x-4)/(x+3) >= 0;
Fa\303\247amos uma lista de valores para que possamos comparar o que acontece com o produto desses dois termos com o quociente desses dois termos.for k from -5 to 8 do
if((k+3) = 0)
then print(`x = `,k,` Produto = `,(k-4)*(k+3), ` Quociente = undefined`):
else print(`x = `,k,` Produto = `,(k-4)*(k+3), ` Quociente = `, evalf((k-4)/(k+3),5));
fi; od;
O que estamos vendo? Temos dois pontos:
1. Quando o produto \303\251 zero, o quociente \303\251 0 ou indefinido. Na verdade \303\251 indefinido quando o quociente \303\251 zero.
2. Apesar de o quocinte e do produto assumirem valores diferentes, eles sempre t\303\252m o mesmo sinal.
Logo, podemos usar o mesmo m\303\251todo de gr\303\241ficos acima - com uma pequena mudan\303\247a. Temos de eleiminar qualquer valor que fa\303\247a o denominador ser igual a zero.
QuoInequal((x-4),(x+3) );solve( (x-4)/(x+3) >= 0 );
Perceba que a extremidade de -3 \303\251 ABERTA e n\303\243o fechada, equanto a extremidade de 4 \303\251 FECHADA!
Outros exemplos:
Ineq := ((x-1)*(x-2))/((x-3)*(x-4)) > 0;
QuoInequal((x-1)*(x-2),(x-3)*(x-4));
print(` `);solve(Ineq );Ineq := ((x-6)*(x+3)) / (x-5) > 0;
QuoInequal( (x-6)*(x+3), (x-5));
print(` `);solve(Ineq );<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8"> 4. Inequa\303\247\303\265es Com Quocientes Mais Complicadas </Font><Equation executable="false" style="Heading 2" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkmbWZyYWNHRiQ2KC1JKG1mZW5jZWRHRiQ2JS1GIzYoLUkjbWlHRiQ2JlEieEYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEoJm1pbnVzO0YnRjcvRj5RJ25vcm1hbEYnLyUmZmVuY2VHRjkvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGOS8lKnN5bW1ldHJpY0dGOS8lKGxhcmdlb3BHRjkvJS5tb3ZhYmxlbGltaXRzR0Y5LyUnYWNjZW50R0Y5LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVi1GNDYmUSJhRidGN0Y6Rj0vJSVzaXplR1EjMTJGJ0Y3RkRGN0ZELUYjNihGM0ZALUY0NiZRImJGJ0Y3RjpGPUY3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGY28vJSliZXZlbGxlZEdGOS1GQTYuUSI+RidGN0ZERkZGSEZKRkxGTkZQRlIvRlVRLDAuMjc3Nzc3OGVtRicvRlhGXHAtRiw2KC1GLzYlLUYjNihGM0ZALUY0NiZRImNGJ0Y3RjpGPUZmbkY3RkRGN0ZELUYjNihGM0ZALUY0NiZRImRGJ0Y3RjpGPUY3RjpGPUZeb0Zhb0Zkb0Zmb0ZmbkY3RkQ=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkmbWZyYWNHRiQ2KC1JKG1mZW5jZWRHRiQ2JS1GIzYoLUkjbWlHRiQ2JlEieEYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEoJm1pbnVzO0YnRjcvRj5RJ25vcm1hbEYnLyUmZmVuY2VHRjkvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGOS8lKnN5bW1ldHJpY0dGOS8lKGxhcmdlb3BHRjkvJS5tb3ZhYmxlbGltaXRzR0Y5LyUnYWNjZW50R0Y5LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVi1GNDYmUSJhRidGN0Y6Rj0vJSVzaXplR1EjMTJGJ0Y3RkRGN0ZELUYjNihGM0ZALUY0NiZRImJGJ0Y3RjpGPUY3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGY28vJSliZXZlbGxlZEdGOS1GQTYuUSI+RidGN0ZERkZGSEZKRkxGTkZQRlIvRlVRLDAuMjc3Nzc3OGVtRicvRlhGXHAtRiw2KC1GLzYlLUYjNihGM0ZALUY0NiZRImNGJ0Y3RjpGPUZmbkY3RkRGN0ZELUYjNihGM0ZALUY0NiZRImRGJ0Y3RjpGPUY3RjpGPUZeb0Zhb0Zkb0Zmb0ZmbkY3RkQ=</Equation><Font size="14" style="Text" bold="true"> </Font></Text-field>
H\303\241 outros problemas envolvendo quocientes que parecem ser bem mais confusos. No entanto, se voc\303\252 simplesmente mover todos os termos para um lado da equa\303\247\303\243o e combinar os termos racionais, o problema ser\303\241 convertido para uma forma mais familiar.
(x-3)/(x+4) > 1;(x-3)/(x+4)- 1 > 0 ;0 < simplify(rhs(%));0 > 1/(x+4);solve(%);
Veja outro exemplo.(x-3)/(x+4) > (x+6)/(x-1);lhs(%) - rhs(%) < 0;simplify(lhs(%)) < 0;solve( 7*(2*x+3)/(x-1)/(x+4) < 0 );<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8"> 5. Gr\303\241ficos de Inequa\303\247\303\265es com Produtos </Font><Font size="14" style="Text" bold="true" foreground="[0,128,128]"> </Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
Voc\303\252 pode visualisar uma equa\303\247\303\243o de valor absoluto observando a interse\303\247\303\243o dos dois gr\303\241ficos.
expr := (x-3)*(x+2): % > 0;
a := -5: b := 5:
s := solve(expr = 0, x): a1 := min( s): b1 := max(s):
plots[display](
plot( expr, x = a..b, thickness = 2),
plots[polygonplot]( [[a,-.5],[a1,-.5],[a1,.5],[a,.5]], color = green),
plots[polygonplot]( [[b1,-.5],[b,-.5],[b,.5],[b1,.5]], color = green)
);
expr := (x-3)*(x+2): % < 0;
a := -5: b := 5:
s := solve(expr = 0, x): a1 := min( s): b1 := max(s):
plots[display](
plot( expr, x = a..b, thickness = 2),
plots[polygonplot]( [[a1,-.5],[b1,-.5],[b1,.5],[a1,.5]], color = green)
);
Estes s\303\243o exemplos de casos mais complicados. expr := (5-x)*(x+2): % < 0;
a := -4: b := 7:
s := solve(expr = 0, x): a1 := min( s): b1 := max(s):
plots[display](
plot( expr, x = a..b, thickness = 2),
plots[polygonplot]( [[a,-.5],[a1,-.5],[a1,.5],[a,.5]], color = green),
plots[polygonplot]( [[b1,-.5],[b,-.5],[b,.5],[b1,.5]], color = green)
);
Perceba que as \303\272nicas partes da solu\303\247\303\243o aqui est\303\243o onde o gr\303\241fico est\303\241 ABAIXO do eixo x.expr := (x+4)*(x+2)*(x-3): % < 0;
a := -5: b := 3.5:
s := solve(expr < 0, x); a1 := -4: a2:= -2: a3:= 3:
plots[display](
plot( expr, x = a..b, thickness = 2),
plots[polygonplot]( [[a,-.5],[a1,-.5],[a1,.5],[a,.5]], color = green),
plots[polygonplot]( [[a2,-.5],[a3,-.5],[a3,.5],[a2,.5]], color = green)
);
Quando o valor absoluto \303\251 MENOR que a constante, devemos procurar pelo V ABAIXO da linha horizontal.expr := (x+4)*(x+2)*(x-3): % < 0;
a := -5: b := 4:
s := solve(expr < 0, x); a1 :=-4: a2:= -2: a3:= 3:
plots[display](
plot( expr, x = a..b, thickness = 2),
plots[polygonplot]( [[a,-.5],[a1,-.5],[a1,.5],[a,.5]], color = green),
plots[polygonplot]( [[a2,-.5],[a3,-.5],[a3,.5],[a2,.5]], color = green)
);
Quando o valor absoluto \303\251 MAIOR que a constante, devemos procurar pelo V ACIMA da linha horizontal.<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8"> 6. Gr\303\241ficos de Inequa\303\247\303\265es com Quocientes </Font><Font size="14" style="Text" bold="true"> </Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
Voc\303\252 tamb\303\251m pode viusualizar um quociente. Nesse caso, estamos procurando onde o gr\303\241fico \303\251 positivo.
expr := (x+4)/(x-3): % > 0; solve(%);
a := -9: b := 11: a1 := -4: a2:= 3:
plots[display](
plot( expr, x = a..b, y = -4..7,thickness = 2, discont = true),
plots[polygonplot]( [[a,-.5],[a1,-.5],[a1,.5],[a,.5]], color = green),
plots[polygonplot]( [[a2,-.5],[b,-.5],[b,.5],[a2,.5]], color = green)
);expr := (x+4)/(x-3): % < 0; solve(%);
a := -9: b := 11: a1 := -4: a2:= 3:
plots[display](
plot( expr, x = a..b, y = -4..7,thickness = 2, discont = true),
plots[polygonplot]( [[a1,-.5],[a2,-.5],[a2,.5],[a1,.5]], color = green)
);expr := ((x+4)*(x-1))/((x-3)*(x+2)): % > 0; solve(%);
a := -9: b := 11: a1 := -4: a2 := -2: a3 := 1: a4 := 3:
plots[display](
plot( expr, x = a..b, y = -4..7,thickness = 2, discont = true),
plots[polygonplot]( [[a,-.5],[a1,-.5],[a1,.5],[a,.5]], color = green),
plots[polygonplot]( [[a2,-.5],[a3,-.5],[a3,.5],[a2,.5]], color = green),
plots[polygonplot]( [[a4,-.5],[b,-.5],[b,.5],[a4,.5]], color = green)
);