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Solve the absolute value inequality for x, |(x+4)/3| > 2 and write your answer using the inequality notation ( < or > symbols).

Sagot :

jacob193

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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