Solve each equation or inequality.∣18−3x∣4\u003c−13|18- 3x | 4 \u003c -13

Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 58

2장, 문제 58

Solve each equation or inequality.
$∣ 18 - 3 x ∣ 4 < - 13$

검증된 단계별 안내

Recognize that the absolute value expression $|18 - 3x|$ represents the distance of $18 - 3x$ from zero on the number line, and absolute values are always greater than or equal to zero.

Note that the inequality is $|18 - 3x| + 4 < -13$. Since $|18 - 3x| \geq 0$, the smallest value of $|18 - 3x| + 4$ is 4, which is already greater than -13.

Because the left side $|18 - 3x| + 4$ can never be less than a negative number like -13, there are no values of $x$ that satisfy this inequality.

Conclude that the inequality has no solution since an absolute value plus a positive number cannot be less than a negative number.

Therefore, the solution set is the empty set, often denoted as $\emptyset$.

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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 where the absolute value of a variable or expression is compared to a number. Understanding how to interpret and solve inequalities like |A| < B or |A| > B is essential, where the solution depends on whether B is positive, zero, or negative.

Properties of Absolute Value

The absolute value of a number represents its distance from zero on the number line and is always non-negative. This means |x| ≥ 0 for any real x, and |x| < 0 has no solution. Recognizing this helps determine if an inequality involving absolute values is possible or not.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable and understanding inequality rules, such as reversing the inequality sign when multiplying or dividing by a negative number. This skill is necessary when breaking down absolute value inequalities into compound inequalities.

Solve each equation or inequality.∣18−3x∣4<−13|18- 3x | 4 < -13