Question

Absolute Value Inequality: Solve: ∣4x∣>8

A) x<2 and x>−2
B) x>2 or x<−2
C) x>2 or x≤−2
D) x>32 or x<−32

Answer 1

The solution to the inequality |4x|>8 is when x is greater than 2 or less than -2. We consider two cases, one where 4x > 8 and another where 4x < -8, and solve each inequality separately.

Step-by-step explanation:

To solve: |4x|>8, we must consider that the absolute value of a number refers to its distance from zero on the number line, without regard to direction. Therefore, |4x|>8 means that 4x is more than 8 units away from zero, on either side of the number line. We can divide the inequality into two separate inequalities without the absolute value:

• 4x > 8

• or

• -4x > 8 (which can be rewritten as 4x < -8)

Solving the first inequality:

1. Divide both sides by 4, giving us x > 2.

Solving the second inequality:

1. Divide both sides by 4, but remember to reverse the inequality sign since we are dividing by a negative number, giving us x < -2.

Thus, the solution to the inequality |4x|>8 is x > 2 or x < -2.