Solve each equation or inequality. | 2x+ 1/3 | +1 \u003c 4

Start by isolating the absolute value expression. Subtract 1 from both sides of the inequality: $$|2x + \frac{1}{3}| + 1 0$$, the solution is $$-B \u003c A \u003c B. $$Apply this to get: $$-3 \u003c 2x + \frac{1}{3} \u003c 3. $$Next, solve the compound inequality by isolating $$x. $$Start with the left part: $$-3 \u003c 2x + \frac{1}{3}. $$Subtract $$\frac{1}{3}$$ from both sides to get $$-3 - \frac{1}{3} \u003c 2x. $$Then, divide both sides of the inequality by 2 to solve for $$x$$: $$\frac{-3 - \frac{1}{3}}{2} \u003c x. $$Repeat the process for the right part of the inequality: $$2x + \frac{1}{3} \u003c 3. $$Subtract $$\frac{1}{3}$$ from both sides and then divide by 2 to find the upper bound for $$x. $$Combine both results to express the solution as an interval.

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 50

2장, 문제 50

Solve each equation or inequality. | 2x+ 1/3 | +1 < 4

검증된 단계별 안내

Start by isolating the absolute value expression. Subtract 1 from both sides of the inequality: $|2x + \frac{1}{3}| + 1 < 4$ becomes $|2x + \frac{1}{3}| < 3$.

Recall that for an inequality of the form $|A| < B$ where $B > 0$, the solution is $-B < A < B$. Apply this to get: $-3 < 2x + \frac{1}{3} < 3$.

Next, solve the compound inequality by isolating $x$. Start with the left part: $-3 < 2x + \frac{1}{3}$. Subtract $\frac{1}{3}$ from both sides to get $-3 - \frac{1}{3} < 2x$.

Then, divide both sides of the inequality by 2 to solve for $x$: $\frac{-3 - \frac{1}{3}}{2} < x$.

Repeat the process for the right part of the inequality: $2x + \frac{1}{3} < 3$. Subtract $\frac{1}{3}$ from both sides and then divide by 2 to find the upper bound for $x$. Combine both results to express the solution as an interval.

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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 or 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 that capture the positive and negative scenarios.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This often involves subtracting or adding constants and dividing by coefficients, ensuring the inequality is in a standard form for applying the absolute value rules.

Solving Linear Inequalities

Once the absolute value is removed, solving the resulting linear inequalities requires applying standard algebraic techniques such as adding, subtracting, multiplying, or dividing both sides by constants, while carefully considering the direction of the inequality when multiplying or dividing by negative numbers.