Question

Describe the solutions of |x| + 2 ≤ 4 and |x| + 2 ≥ 4 using inequalities that don’t involve absolute value. a) -6 ≤ x ≤ 2 , -6 ≥ x ≥ 2

b) -6 ≥ x ≤ 2 , -6 ≤ x ≥ 2
c) -2 ≤ x ≤ 6 , -2 ≥ x ≥ 6
d) -2 ≥ x ≤ 6 , -2 ≤ x ≥ 6

Answer 1

To solve the inequalities |x| + 2 ≤ 4 and |x| + 2 ≥ 4, we rewrite them without absolute value signs and solve separately. The solutions are -2 ≤ x ≤ 2.

Step-by-step explanation:

To solve the inequality |x| + 2 ≤ 4, we can rewrite it as x + 2 ≤ 4 and -x + 2 ≤ 4. Solving these two inequalities separately, we subtract 2 from both sides to get x ≤ 2 and -x ≤ 2, which can be rewritten as x ≥ -2. Combining these two solutions, we have -2 ≤ x ≤ 2.

To solve the inequality |x| + 2 ≥ 4, we can rewrite it as x + 2 ≥ 4 and -x + 2 ≥ 4. Solving these two inequalities separately, we subtract 2 from both sides to get x ≥ 2 and -x ≥ 2, which can be rewritten as x ≤ -2. Combining these two solutions, we have -2 ≤ x ≤ 2.