In Exercises 51–58, solve each compound inequality. - 11 \u003c 2x - 1 ≤ - 5

Start by writing the compound inequality as two separate inequalities combined: $$11 \u003c 2x - 1$$ and $$2x - 1 \leq -5. $$Solve the first inequality $$11 \u003c 2x - 1 by $$adding 1 to both sides to isolate the term with $$x$$: $$11 + 1 \u003c 2x$$ which simplifies to $$12 \u003c 2x. $$Next, divide both sides of $$12 \u003c 2x by 2 to $$solve for $$x$$: $$\frac{12}{2} \u003c x$$ which simplifies to $$6 \u003c x. $$Now solve the second inequality $$2x - 1 \leq -5 by $$adding 1 to both sides: $$2x \leq -5 + 1$$ which simplifies to $$2x \leq -4. $$Divide both sides of $$2x \leq -4 by 2 to $$solve for $$x$$: $$x \leq \frac{-4}{2}$$ which simplifies to $$x \leq -2.$$

Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 55

2장, 문제 55

In Exercises 51–58, solve each compound inequality. - 11 < 2x - 1 ≤ - 5

검증된 단계별 안내

Start by writing the compound inequality as two separate inequalities combined: $11 < 2x - 1$ and $2x - 1 \leq -5$.

Solve the first inequality $11 < 2x - 1$ by adding 1 to both sides to isolate the term with $x$: $11 + 1 < 2x$ which simplifies to $12 < 2x$.

Next, divide both sides of $12 < 2x$ by 2 to solve for $x$: $\frac{12}{2} < x$ which simplifies to $6 < x$.

Now solve the second inequality $2x - 1 \leq -5$ by adding 1 to both sides: $2x \leq -5 + 1$ which simplifies to $2x \leq -4$.

Divide both sides of $2x \leq -4$ by 2 to solve for $x$: $x \leq \frac{-4}{2}$ which simplifies to $x \leq -2$.

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주요 개념

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

Compound Inequalities

Compound inequalities involve two inequalities combined into one statement, often connected by 'and' or 'or'. Solving them requires finding values that satisfy both inequalities simultaneously, typically represented as a range or union of intervals.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side by performing inverse operations, similar to solving equations, but remembering to reverse the inequality sign when multiplying or dividing by a negative number.

Interval Notation and Graphing Solutions

After solving inequalities, solutions are often expressed in interval notation, which concisely describes all values that satisfy the inequality. Graphing these intervals on a number line helps visualize the solution set clearly.