Use the table to solve each inequality. - 3 \u003c 2x - 5 ≤ 3

Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 106

2장, 문제 106

Use the table to solve each inequality. - 3 < 2x - 5 ≤ 3

검증된 단계별 안내

Start with the compound inequality: \(3 < 2x - 5 \leq 3\).

Break the compound inequality into two separate inequalities: 1) \(3 < 2x - 5\) 2) \(2x - 5 \leq 3\).

Solve the first inequality \(3 < 2x - 5\) by adding 5 to both sides: \(3 + 5 < 2x\), which simplifies to \(8 < 2x\). Then divide both sides by 2 to isolate \(x\): \(4 < x\).

Solve the second inequality \(2x - 5 \leq 3\) by adding 5 to both sides: \(2x \leq 8\). Then divide both sides by 2: \(x \leq 4\).

Combine the two results to find the solution to the compound inequality: \(4 < x \leq 4\). Since this is a contradiction (no number is both greater than 4 and less than or equal to 4), check the table values for \(x\) and \(y = 2x - 13\) to verify if any \(x\) satisfies the original inequality.

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주요 개념

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

Solving Compound Inequalities

A compound inequality involves two inequalities joined by 'and' or 'or'. To solve, treat each inequality separately and find the intersection (for 'and') or union (for 'or') of their solution sets. For example, solving 3 < 2x - 5 ≤ 3 requires isolating x in both inequalities and combining the results.

Using Tables to Solve Inequalities

Tables can help solve inequalities by providing values of expressions for specific x-values. By comparing the expression values to the inequality bounds, you can identify which x-values satisfy the inequality. This method is useful when direct algebraic manipulation is complex or when verifying solutions.

Linear Functions and Their Graphs

A linear function like y = 2x - 13 produces a straight line when graphed. Understanding how changes in x affect y helps solve inequalities involving linear expressions. The table shows corresponding y-values for given x-values, illustrating the function's behavior and aiding in solving inequalities.