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Sagot :
Answer:
Either [tex]x > 2[/tex], or [tex]x < -10[/tex].
Step-by-step explanation:
Make use of the following property of absolute value inequalities: given a positive number [tex]p[/tex] ([tex]p > 0[/tex]) and some algebraic expression [tex]X[/tex], the following are equivalent:
- [tex]|X| > p[/tex], and
- either [tex]X < (-p)[/tex] or [tex]X > p[/tex].
In this question, [tex]| (x + 4) / 3| > 2[/tex] would be equivalent to [tex]((x + 4) / 3) > 2[/tex] or [tex]((x + 4) / 3) < -2[/tex]. Simplify to separate [tex]x[/tex]:
[tex]\displaystyle \frac{x + 4}{3} > 2 \quad \text{or} \quad \frac{x + 4}{3} < -2[/tex].
[tex]x + 4 > 6 \quad \text{or} \quad x + 4 < -6[/tex].
[tex]x > 2 \quad \text{or} \quad x < -10[/tex].
In other words, the given inequality is satisfied if and only if either [tex]x > 2[/tex] or [tex]x < -10[/tex].
The final solution is x < -10 or x > 2.
To solve the inequality, we need to break it down into two separate inequalities:
1. (x+4)/3 > 2
2. (x+4)/3 < -2
Solve (x+4)/3 > 2:
- Multiply both sides by 3:
(x+4)/3 *3 > 2 * 3
(x+4) > 6
- Subtract 4 from both sides:
(x+4) - 4 > 6 - 4
x > 2
Solve (x+4)/3 < -2:
- Multiply both sides by 3:
(x+4)/3 *3 < -2 * 3
(x+4) < -6
- Subtract 4 from both sides:
(x+4) - 4 < -6 - 4
x < -10
Thus, the solution to the inequality |(x+4)/3| > 2 is x < -10 or x > 2.
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