Question
Find the solution to the absolute value inequality −3|x−4|<6.(1 point)
Responses
no solution
no solution
(2,6)
left parenthesis 2 comma 6 right parenthesis
(−∞,∞)
left parenthesis negative infinity comma infinity right parenthesis
(−∞,2)∪(6,∞)
Responses
no solution
no solution
(2,6)
left parenthesis 2 comma 6 right parenthesis
(−∞,∞)
left parenthesis negative infinity comma infinity right parenthesis
(−∞,2)∪(6,∞)
Answers
GPT-4o mini
To solve the inequality \(-3|x−4| < 6\), we can start by dividing both sides by -3. Remember, when we divide by a negative number, we flip the inequality sign.
\[
|x−4| > -2
\]
Since the absolute value \(|x−4|\) is always non-negative, it is always greater than \(-2\). Therefore, this inequality is satisfied for all \(x\).
Thus, the solution to the inequality is:
\[
(-\infty, \infty)
\]
The correct response is \((-∞,∞)\).
\[
|x−4| > -2
\]
Since the absolute value \(|x−4|\) is always non-negative, it is always greater than \(-2\). Therefore, this inequality is satisfied for all \(x\).
Thus, the solution to the inequality is:
\[
(-\infty, \infty)
\]
The correct response is \((-∞,∞)\).