Questions: U=x ∈ Z ; 3x-2 ≤ 21

U=x ∈ Z ; 3x-2 ≤ 21
Transcript text: $U=\{x \in \mathbb{Z} ; 3|x-2| \leq 21\}$
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Solution

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Solution Steps

To solve the inequality \(3|x-2| \leq 21\), we first divide both sides by 3 to simplify it to \(|x-2| \leq 7\). This absolute value inequality can be split into two separate inequalities: \(x-2 \leq 7\) and \(x-2 \geq -7\). Solving these inequalities will give us the range of values for \(x\).

Step 1: Simplificação da Desigualdade

Começamos com a desigualdade \(3|x-2| \leq 21\). Dividindo ambos os lados por 3, obtemos: \[ |x-2| \leq 7 \]

Step 2: Resolução da Desigualdade Absoluta

A desigualdade absoluta pode ser dividida em duas desigualdades: \[ -7 \leq x - 2 \leq 7 \] Resolvendo cada parte, temos:

  1. \(x - 2 \leq 7\) resulta em \(x \leq 9\)
  2. \(x - 2 \geq -7\) resulta em \(x \geq -5\)
Step 3: Intervalo de Solução

Combinando as duas soluções, obtemos o intervalo: \[ -5 \leq x \leq 9 \]

Final Answer

A solução para a desigualdade é: \[ \boxed{-5 \leq x \leq 9} \]

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