Question

solve the absolute value inequality. complete parts a) through d).

\\|3x - 2| \ge 7\\

a) if necessary, write the inequality in standard form. select the correct choice and, if necessary, fill in the answer boxes to complete your choice.

a. the absolute value inequality \\(|3x - 2| \ge 7\\) is not in standard form, but can be written in standard form as \\(\boxed{\phantom{x}} \ge \boxed{\phantom{x}}\\)
(simplify your answers.)

b. the absolute value inequality \\(|3x - 2| \ge 7\\) is already in standard form.

b) rewrite the inequality from part a) without using absolute value bars. select the correct choice and fill in the answer boxes to complete your choice.
(simplify your answers. do not evaluate. use ascending order.)

a. the inequality can be rewritten as the three-part inequality \\(\boxed{\phantom{x}} \le \boxed{\phantom{x}} \le \boxed{\phantom{x}}\\)

b. the inequality can be rewritten as the compound inequality \\(\boxed{\phantom{x}} \le \boxed{\phantom{x}}\\) or \\(\boxed{\phantom{x}} \ge \boxed{\phantom{x}}\\)

c. the inequality can be rewritten as the inequality \\(\boxed{\phantom{x}} \le \boxed{\phantom{x}}\\)

Explanation:

Identify the standard form of the absolute value inequality

$$ |3x - 2| \ge 7 $$

The inequality is already in the standard form \(|ax + b| \ge c\).

Rewrite the inequality without absolute value bars

Using the property \(|u| \ge k \iff u \le -k \text{ or } u \ge k\) for \(k > 0\):

$$ 3x - 2 \le -7 \quad \text{or} \quad 3x - 2 \ge 7 $$

This corresponds to option B, written in ascending order of the constant bounds.

Answer:

Question a

• A. The absolute value inequality \(|3x - 2| \ge 7\) is not in standard form, but can be written in standard form as \(\Box \ge \Box\).
• B. The absolute value inequality \(|3x - 2| \ge 7\) is already in standard form. (Correct answer)

Question b

• A. The inequality can be rewritten as the three-part inequality \(\Box \le \Box \le \Box\).
• B. The inequality can be rewritten as the compound inequality \(3x - 2 \le -7\) or \(3x - 2 \ge 7\). (Correct answer)
• C. The inequality can be rewritten as the inequality \(\Box \le \Box\).