Solve each equation or inequality.∣7x+8∣−6\u003e−3|7x+8| - 6 \u003e -3

Start by isolating the absolute value expression on one side of the inequality. Add 6 to both sides to get: $$|7x + 8| \u003e -3 + 6. $$Simplify the right side of the inequality: $$|7x + 8| \u003e 3. $$Recall that the absolute value $$|A| \u003e B ($$where $$B \u003e 0) $$means that either $$A \u003e B or$$ $$A 3 or$$ $$7x + 8 3$$, subtract 8 from both sides and then divide by 7. For $$7x + 8 \u003c -3$$, subtract 8 from both sides and then divide by 7. Write the solution as the union of the two solution sets found in the previous step, representing all $$x$$ values that satisfy the original inequality.

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 118

2장, 문제 118

Solve each equation or inequality.
$∣ 7 x + 8 ∣ - 6 > - 3$

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Start by isolating the absolute value expression on one side of the inequality. Add 6 to both sides to get: $|7x + 8| > -3 + 6$.

Simplify the right side of the inequality: $|7x + 8| > 3$.

Recall that the absolute value $|A| > B$ (where $B > 0$) means that either $A > B$ or $A < -B$. So, set up two inequalities: $7x + 8 > 3$ or $7x + 8 < -3$.

Solve each inequality separately. For $7x + 8 > 3$, subtract 8 from both sides and then divide by 7. For $7x + 8 < -3$, subtract 8 from both sides and then divide by 7.

Write the solution as the union of the two solution sets found in the previous step, representing all $x$ values that satisfy the original inequality.

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주요 개념

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Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value bars and require considering the distance from zero on the number line. To solve, rewrite the inequality without absolute value by splitting it into two cases based on the definition of absolute value.

Properties of Inequalities

When solving inequalities, operations like adding, subtracting, multiplying, or dividing both sides must preserve the inequality's direction, except when multiplying or dividing by a negative number, which reverses it. Understanding these rules is essential for correctly manipulating and solving inequalities.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value expression on one side of the inequality. This step simplifies the problem and allows you to apply the definition of absolute value to split the inequality into two separate inequalities.

Solve each equation or inequality.∣7x+8∣−6>−3|7x+8| - 6 > -3