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Chapter 1: Problem 44

Solve the inequality, and write the solution set in interval notation. \(15<|-2 d-3|+6\)

Short Answer

Expert verified
(-鈭�, -6) 鈭� (3, 鈭�)

Step by step solution

01

Isolate the Absolute Value Expression

Start by isolating the absolute value expression Subtract 6 from both sides of the inequality: 15 - 6 < |-2d - 3| 9 < |-2d - 3|
02

Set Up Two Inequalities

Since |a| > b means a > b or a < -b, this translates to: -2d - 3 > 9 or -2d - 3 < -9
03

Solve the First Inequality

-2d - 3 > 9 Add 3 to both sides: -2d > 12 Divide both sides by -2 and reverse the inequality sign: d < -6
04

Solve the Second Inequality

-2d - 3 < -9 Add 3 to both sides: -2d < -6 Divide both sides by -2 and reverse the inequality sign: d > 3
05

Combine the Solutions

Combine the solutions from Step 3 and Step 4: d < -6 or d > 3
06

Write the Solution Set in Interval Notation

Convert the solution to interval notation: (-鈭�, -6) 鈭� (3, 鈭�)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

absolute value inequalities
Absolute value inequalities involve expressions within absolute value bars (| |). In these cases, an absolute value represents the distance of a number from zero on the number line, always resulting in a non-negative number. For example, \( |-2| = 2 \)' and \( |3| = 3 \).
For inequalities, the key is to isolate the absolute value expression first. This helps break the problem into manageable parts.
When you encounter something like \( |a| > b \), two possibilities arise:
  • \

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Most popular questions from this chapter

Use the relationship between temperature in Celsius and temperature in Fahrenheit. \(C=\frac{5}{9}(F-32)\) Body temperature is usually maintained between \(36.5^{\circ} \mathrm{C}\) and \(37.5^{\circ} \mathrm{C}\), inclusive. Determine the corresponding range of temperature in Fahrenheit.

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