Which inequality has solution set (-∞, ∞)? A. (x-3)²≥0 B. (5x-6)²≤0 C. (6x+4)²>0 D. (8x+7)²<0

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1. Recall that for any real number expression, a square of that expression, such as \( (expression)^2 \), is always greater than or equal to zero because squaring any real number cannot produce a negative result.
2. Analyze option A: \( (x-3)^2 \geq 0 \). Since squares are always non-negative, this inequality holds for all real values of \(x\), so its solution set is \((-\infty, \infty)\).
3. Analyze option B: \( (5x-6)^2 \leq 0 \). Since squares are always non-negative, the only way this inequality holds is when \( (5x-6)^2 = 0 \), which happens at a single value of \(x\). So the solution set is not all real numbers.
4. Analyze option C: \( (6x+4)^2 > 0 \). This inequality is true for all \(x\) except where \( (6x+4)^2 = 0 \), which is at one specific \(x\) value. So the solution set excludes one point, not all real numbers.
5. Analyze option D: \( (8x+7)^2 < 0 \). Since squares cannot be negative, this inequality has no solution, so the solution set is empty.

Key Concepts

Properties of Squares and Non-negativity

The square of any real number is always non-negative, meaning it is either zero or positive. This property helps determine the possible values of expressions like (x - 3)^2, which can never be negative.

Solving Inequalities Involving Squares

Inequalities with squared expressions require understanding when the expression equals zero or is positive/negative. For example, (expression)^2 ≥ 0 is always true, while (expression)^2 < 0 has no real solutions since squares can't be negative.

Interpreting Solution Sets of Inequalities

The solution set (-∞, ∞) means all real numbers satisfy the inequality. Recognizing which inequalities hold for all x involves analyzing the inequality's nature, such as whether it can ever be false or has restrictions.