Solve each equation or inequality.∣8−5x∣≥2|8-5x| ≥ 2

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 114

2장, 문제 114

Solve each equation or inequality.
$∣ 8 - 5 x ∣ \geq 2$

검증된 단계별 안내

Recognize that the inequality involves an absolute value expression: $|8 - 5x| \geq 2$. Recall that for any expression $A$, $|A| \geq k$ means $A \geq k$ or $A \leq -k$ when $k > 0$.

Set up two separate inequalities based on the definition of absolute value inequalities: 1) $8 - 5x \geq 2$ 2) $8 - 5x \leq -2$

Solve the first inequality $8 - 5x \geq 2$ by isolating $x$: Subtract 8 from both sides: $-5x \geq 2 - 8$ Simplify the right side: $-5x \geq -6$ Divide both sides by $-5$, remembering to reverse the inequality sign because you are dividing by a negative number: $x \leq \frac{-6}{-5}$

Solve the second inequality $8 - 5x \leq -2$ similarly: Subtract 8 from both sides: $-5x \leq -2 - 8$ Simplify the right side: $-5x \leq -10$ Divide both sides by $-5$, reversing the inequality sign: $x \geq \frac{-10}{-5}$

Combine the two solution sets from steps 3 and 4 to express the final solution as a union of intervals where $x$ satisfies either inequality.

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주요 개념

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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 includes values where A ≤ -B or A ≥ B, reflecting the two-sided nature of absolute value.

Solving Linear Inequalities

Solving linear inequalities requires isolating the variable while maintaining inequality direction. When multiplying or dividing by a negative number, the inequality sign must be reversed to preserve the inequality's truth.

Splitting Absolute Value Expressions

To solve |expression| ≥ number, split the inequality into two cases: one where the expression is greater than or equal to the number, and one where it is less than or equal to the negative of that number. Solve each case separately.