Question

What are the solutions to the absolute value inequality |4x| - 2 ≥14?

Answer 1

To solve the absolute value inequality |4x| - 2 ≥ 14, we need to isolate the absolute value expression and find the values of x that satisfy the inequality.

First, let's isolate the absolute value expression by adding 2 to both sides of the inequality:

|4x| - 2 + 2 ≥ 14 + 2

|4x| ≥ 16

Now, we can split the inequality into two separate cases, considering both the positive and negative values of 4x:

1) When 4x is positive, we have:

4x ≥ 16

Divide both sides by 4:

x ≥ 16/4

x ≥ 4

2) When 4x is negative, we have:

-4x ≥ 16

Divide both sides by -4, and remember to reverse the inequality:

x ≤ -16/4

x ≤ -4

Thus, the solutions to the absolute value inequality |4x| - 2 ≥ 14 are x ≥ 4 or x ≤ -4.

Answer 2

The solutions to the absolute value inequality |4x| - 2 ≥ 14 are found by first isolating the absolute value and then creating two separate inequalities. Solving both inequalities results in the solution set: x ≥ 4 or x ≤ -4.

Step-by-step explanation:

The solutions to the absolute value inequality |4x| - 2 ≥ 14 can be found by isolating the absolute value expression and then considering the two possible cases for the absolute value. First, we add 2 to both sides to get |4x| ≥ 16. Then we split it into two separate inequalities: 4x ≥ 16 and 4x ≤ -16 (since |a| ≥ b implies a ≥ b or a ≤ -b).

For the first inequality 4x ≥ 16, we divide by 4 to find x ≥ 4. For the second inequality 4x ≤ -16, we also divide by 4 to find x ≤ -4.

Thus, the solution set is x ≥ 4 or x ≤ -4.