Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

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

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.

Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.