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Difficulty: MediumAbsolute Value Equations and Inequalities

What is the complete solution set for the inequality ∣7−3n∣<11|7 - 3n| < 11?

  1. A
    n<−43n < -\frac{4}{3} or n>6n > 6
  2. −43<n<6-\frac{4}{3} < n < 6Answer
  3. C
    −6<n<43-6 < n < \frac{4}{3}
  4. D
    n>−43n > -\frac{4}{3}
  5. E
    n<−43n < -\frac{4}{3}

Answer

−43<n<6-\frac{4}{3} < n < 6
To solve ∣7−3n∣<11|7 - 3n| < 11, rewrite it as the compound inequality −11<7−3n<11-11 < 7 - 3n < 11. Subtracting 77 from all parts yields −18<−3n<4-18 < -3n < 4. Finally, dividing by −3-3 and reversing the inequality signs gives 6>n>−436 > n > -\frac{4}{3}, which simplifies to the interval −43<n<6-\frac{4}{3} < n < 6. This matches the correct option.

Step-by-Step Solution

1
Express the absolute value inequality as a compound inequality.
−11<7−3n<11-11 < 7 - 3n < 11
An absolute value inequality of the form ∣u∣<c|u| < c is equivalent to the compound inequality −c<u<c-c < u < c.
2
Subtract 77 from all three parts of the inequality.
−18<−3n<4-18 < -3n < 4
To isolate the term containing nn, we subtract 77 from all parts of the inequality.
3
Divide all three parts by −3-3 and reverse the inequality signs.
6>n>−436 > n > -\frac{4}{3}, which is equivalent to −43<n<6-\frac{4}{3} < n < 6
Dividing an inequality by a negative number requires reversing the direction of the inequality signs to maintain a true statement.

Key Concept

Absolute Value Equations and Inequalities
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Topics

Absolute Value Equations and Inequalities

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