Solve each equation or inequality. | 5x + 1/2 | -2 \u003c 5

Start by isolating the absolute value expression. Add 2 to both sides of the inequality to get: $$| 5x + \frac{1}{2} | 0$$, the solution is $$-B \u003c A \u003c B. $$Apply this to get: $$-7 \u003c 5x + \frac{1}{2} \u003c 7. $$Next, solve the compound inequality by subtracting $$\frac{1}{2}$$ from all parts: $$-7 - \frac{1}{2} \u003c 5x \u003c 7 - \frac{1}{2}. $$Simplify the expressions on both sides: $$-\frac{15}{2} \u003c 5x \u003c \frac{13}{2}. $$Finally, divide all parts of the inequality by 5 to isolate $$x$$: $$-\frac{15}{10} \u003c x \u003c \frac{13}{10}. $$Simplify the fractions if possible.

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 49

2장, 문제 49

Solve each equation or inequality. | 5x + 1/2 | -2 < 5

검증된 단계별 안내

Start by isolating the absolute value expression. Add 2 to both sides of the inequality to get: $| 5x + \frac{1}{2} | < 7$.

Recall that for an inequality of the form $|A| < B$, where $B > 0$, the solution is $-B < A < B$. Apply this to get: $-7 < 5x + \frac{1}{2} < 7$.

Next, solve the compound inequality by subtracting $\frac{1}{2}$ from all parts: $-7 - \frac{1}{2} < 5x < 7 - \frac{1}{2}$.

Simplify the expressions on both sides: $-\frac{15}{2} < 5x < \frac{13}{2}$.

Finally, divide all parts of the inequality by 5 to isolate $x$: $-\frac{15}{10} < x < \frac{13}{10}$. Simplify the fractions if possible.

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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 expression is compared to a number. To solve them, you consider the definition of absolute value as distance from zero, leading to two separate inequalities to solve.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This step simplifies the problem and allows you to apply the rules for solving absolute value inequalities correctly.

Solving Linear Inequalities

After splitting the absolute value inequality into two linear inequalities, solve each by isolating the variable. This involves standard algebraic techniques such as adding, subtracting, multiplying, or dividing both sides while considering inequality direction.