What is the solution for x in the absolute value inequality |2x + 7| ≤ 0?

To solve the absolute value inequality |2x + 7| ≤ 0, it is essential to understand the properties of absolute values. The expression |a| represents the distance of 'a' from zero on the number line, which is always non-negative. Therefore, |2x + 7| can equal zero only if the expression inside the absolute value is also zero.

Setting the expression inside the absolute value to zero gives us the equation:

2x + 7 = 0

When solving for x, we subtract 7 from both sides:

2x = -7

Next, we divide both sides by 2:

x = -7/2

Since an absolute value cannot be less than zero, the inequality |2x + 7| ≤ 0 implies that |2x + 7| can only equal zero, which leads us to the single solution x = -7/2.

In this context, there are no values of x that would satisfy |2x + 7| being negative, hence the solution x = -7/2 correctly reflects that the only point where the expression meets the condition of the inequality is at that specific x-value. Other options suggest values or scenarios without a valid