Solve each equation or inequality.∣1.5x−14∣\u003c0 | 1.5x - 14| \u003c 0

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 62

2장, 문제 62

Solve each equation or inequality.
$∣ 1.5 x - 14 ∣ < 0$

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Understand the inequality $ |1.5x - 14| < 0 $. The absolute value $ |A| $ represents the distance of $ A $ from zero, which is always $ \geq 0 $.

Recall that the absolute value of any real number is never negative, so $ |1.5x - 14| < 0 $ means we are looking for values of $ x $ where the absolute value is less than zero.

Since absolute values cannot be negative, there are no real numbers $ x $ that satisfy $ |1.5x - 14| < 0 $.

Therefore, the solution set is empty, meaning no solution exists for this inequality.

In summary, inequalities of the form $ |expression| < 0 $ have no solutions because absolute values are always $ \geq 0 $.

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주요 개념

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

Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value bars, representing distance from zero. For inequalities like |A| < B, the solution requires B to be positive, and the expression A must lie between -B and B. Understanding how to interpret and manipulate these inequalities is essential for solving them.

Properties of Absolute Value

The absolute value of a number is always non-negative, meaning |x| ≥ 0 for any real x. This property implies that an absolute value expression cannot be less than zero, which is crucial when determining if an inequality has solutions or not.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side using algebraic operations while maintaining inequality direction. When combined with absolute value expressions, it requires careful consideration of the inequality's conditions to find valid solution sets or conclude no solution exists.

Solve each equation or inequality.∣1.5x−14∣<0 | 1.5x - 14| < 0