Solve each equation or inequality.∣7x−3∣\u003e4|7x-3| \u003e 4

Recognize that the inequality involves an absolute value expression: $$|7x - 3| \u003e 4. $$Recall that for any expression $$A$$, the inequality $$|A| \u003e c ($$where $$c \u003e 0) $$means $$A \u003e c or$$ $$A 4 2$$) $$7x - 3 4 by $$adding 3 to both sides: $$7x \u003e 7$$ Then divide both sides by 7: $$x \u003e 1$$ Solve the second inequality $$7x - 3 1$$

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 115

2장, 문제 115

Solve each equation or inequality.
$∣ 7 x - 3 ∣ > 4$

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Recognize that the inequality involves an absolute value expression: $|7x - 3| > 4$. Recall that for any expression $A$, the inequality $|A| > c$ (where $c > 0$) means $A > c$ or $A < -c$.

Set up two separate inequalities based on the definition of absolute value inequalities: 1) $7x - 3 > 4$ 2) $7x - 3 < -4$

Solve the first inequality $7x - 3 > 4$ by adding 3 to both sides: $7x > 7$ Then divide both sides by 7: $x > 1$

Solve the second inequality $7x - 3 < -4$ by adding 3 to both sides: $7x < -1$ Then divide both sides by 7: $x < -\frac{1}{7}$

Combine the two solution sets to express the final solution: $x < -\frac{1}{7}$ or $x > 1$

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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 distance of a number from zero is compared to another value. For |A| > B, the solution splits into two cases: A > B or A < -B, because the absolute value measures magnitude regardless of sign.

Solving Linear Inequalities

Solving linear inequalities requires isolating the variable on one side while maintaining the inequality's direction. When multiplying or dividing by a negative number, the inequality sign reverses. Solutions are often expressed as intervals or compound inequalities.

Compound Inequalities

Compound inequalities combine two inequalities connected by 'and' or 'or'. For absolute value inequalities like |7x - 3| > 4, the solution is a union of two inequalities, representing values that satisfy either condition, reflecting the distance concept.

Solve each equation or inequality.∣7x−3∣>4|7x-3| > 4