10.2: Resolvendo e representando graficamente desigualdades e escrevendo respostas em notação de intervalo

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Page ID: 170364

Victoria Dominguez, Cristian Martinez, and Sanaa Saykali
Citrus College via ASCCC Open Educational Resources Initiative

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Para resolver e representar graficamente as desigualdades:

Resolva a desigualdade usando as Propriedades das desigualdades da seção anterior.
Faça um gráfico do conjunto de soluções em uma reta numérica.
Escreva o conjunto de soluções em notação de intervalo.

Exemplo Template:index

Resolva a desigualdade, represente graficamente o conjunto de soluções em uma reta numérica e mostre o conjunto de soluções em notação de intervalo:

$−1 ≤ 2x − 5 < 7$
$x^2 + 7x + 10 < 0$
$−6 < x − 2 < 4$

Solução

$\begin{array} &&−1 ≤ 2x − 5 < 7 &\text{Example problem} \\ &−1 + 5 ≤ 2x − 5 + 5 < 7 + 5 &\text{The goal is to isolate the variable \(x$, então comece adicionando$5$ todas as três regiões da desigualdade.}\\ &4 ≤ 2x < 12 &\ text {Simplifique.}\\ &\ dfrac {4} {2} ≤ 2x^2 <\ dfrac {4} {2} &\ text {Divida tudo por$2$ para isolar a variável$x$.}\\ &2 ≤ x < 6 &\ text {Resposta final escrita em inequality/solution set form.}\\ & [2, 6) &\ text {Resposta final escrita em notação de intervalo (consulte a seção sobre Notação de intervalo para obter mais detalhes)}\ end {array}\)

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$\begin{array} &&x^2 + 7x + 10 < 0 &\text{Example problem} \\ &(x + 5)(x + 2) < 0 &\text{Factor the polynomial.} \\ &(x + 5)(x + 2) < 0 &\text{The product must be less than \(0$, o que significa que se$(x + 5) > 0$, então$(x + 2) < 0$. Da mesma forma$(x + 5) < 0$, se, então$(x + 2) > 0$.}\\ & (x + 5) > 0 (x + 2) < 0 &\ text {Find the intersection of each of these inequalities.}\\ &x > −5 x < −2 &\ text {Encontre a interseção de cada uma dessas desigualdades.} \ end {matriz}\)

$\begin{array} &&\;\;\;−5 < x < −2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{Final answer written in inequality/solution set form.} \\ &\;\;\;(−5, −2) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{Final answer written in interval notation (see section on Interval Notation for more details).} \end{array}$

$\begin{array}&&−6 < x − 2 ≤ 4 &\text{Example problem} \\ &−6 + 2 < x − 2 + 2 ≤ 4 + 2 &\text{The goal is to isolate the variable \(x$, então comece adicionando$2$ todas as três regiões da desigualdade.}\\ &−4 < x ≤ 6 &\ text {Resposta final escrita no formato de desigualdade/conjunto de soluções.}\\ & (−4, 6] &\ text {Resposta final escrita em notação de intervalo (consulte a seção Notação de intervalo para obter mais detalhes).} \ end {matriz}\)

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Exercício Template:index

Resolva as desigualdades, represente graficamente os conjuntos de soluções em uma reta numérica e mostre os conjuntos de soluções em notação de intervalo:

$0 ≤ x + 1 ≤ 4$
$0 < 2(x − 1) ≤ 4$
$6 < 2(x − 1) < 12$
$x^2 − 6x − 16 < 0$
$2x^2 − x − 15 > 0$