Answered

Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

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.

We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.