Which inequality has solution set (-∞, ∞)? A. (x-3)2≥0 B. (5x-6)2≤0 C. (6x+4)2\u003e0 D. (8x+7)2\u003c0

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 55

2장, 문제 55

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

검증된 단계별 안내

Recall that for any real number expression squared, such as \(a^2\), the value is always greater than or equal to zero, i.e., \(a^2 \geq 0\) for all real \(a\).

Analyze each inequality option to determine the set of \(x\) values that satisfy it:

Option A: \((x-3)^2 \geq 0\) means the square of \((x-3)\) is greater than or equal to zero. Since squares are never negative, this inequality holds for all real \(x\).

Option B: \((5x-6)^2 \leq 0\) means the square of \((5x-6)\) is less than or equal to zero. Since squares are always nonnegative, the only way this is true is when \((5x-6)^2 = 0\), which happens at a single value of \(x\).

Options C and D involve strict inequalities with squares: \((6x+4)^2 > 0\) and \((8x+7)^2 < 0\). Since squares are never negative, \((8x+7)^2 < 0\) has no solution, and \((6x+4)^2 > 0\) is true for all \(x\) except where the expression inside the square is zero.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Properties of Squares of Real Numbers

The square of any real number is always non-negative, meaning it is either zero or positive. This property is fundamental when analyzing inequalities involving squared expressions, as it restricts the possible values the expression can take.

Solving Inequalities Involving Squares

Inequalities with squared terms often require understanding when the expression equals zero or is positive/negative. Since squares are never negative, inequalities like (expression) < 0 have no real solutions, while (expression) ≥ 0 are true for all real numbers.

Solution Sets and Interval Notation

The solution set of an inequality is the set of all values that satisfy it, often expressed in interval notation. For example, (-∞, ∞) means all real numbers satisfy the inequality, which occurs when the inequality is always true regardless of x.

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