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Chapter 1: Problem 38

Solve the inequality, and write the solution set in interval notation. \(2|7-y|+1<17\)

Short Answer

Expert verified
The solution set is (-1, 15)

Step by step solution

01

- Isolate the Absolute Value Expression

Subtract 1 from both sides of the inequality:\(2|7-y| + 1 - 1 < 17 - 1\)This simplifies to:\(2|7-y| < 16\)
02

- Divide by the Coefficient of the Absolute Value

Divide both sides of the inequality by 2 to isolate the absolute value expression:\(\frac{2|7-y|}{2} < \frac{16}{2}\)This simplifies to:\(|7-y| < 8\)
03

- Set Up Two Separate Inequalities

Since the absolute value of a number less than 8 means that the number is between -8 and 8, set up the following two inequalities:\(-8 < 7-y < 8\)
04

- Solve Each Inequality

First inequality: \(-8 < 7 - y\)1. Subtract 7 from both sides: \(-8 - 7 < - y\)2. Simplify: \(-15 < -y\)3. Multiply both sides by -1 and reverse the inequality sign:\(15 > y\)This simplifies to:\(y < 15\)Second inequality: \(7 - y < 8\)1. Subtract 7 from both sides: \(7 - 7 - y < 8 - 7\)2. Simplify: \(-y < 1\)3. Multiply both sides by -1 and reverse the inequality sign:\(y > -1\)This simplifies to:\(y > -1\)
05

- Combine the Inequalities and Write the Solution in Interval Notation

Combine the two inequalities \(y > -1\) and \(y < 15\) to get:\(-1 < y < 15\)Express the solution set in interval notation:\((-1, 15)\)

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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 an absolute value expression, such as \(|x| < a\), where \(a\) is a positive number. The absolute value refers to the distance of a number from zero on the number line, regardless of direction. Therefore, if \(|x| < a\), it means that \(x\) must fall within the range of \(-a\) and \(a\). For example, for \(|7-y| < 8\), this can be split into two separate inequalities: \(7-y < 8\) and \(-8 < 7-y\). This allows us to handle the absolute value as two regular inequalities that we can solve separately.
interval notation
Interval notation is a shorthand used to describe the set of solutions for inequalities. It provides a concise way to show the range of values that satisfy the given condition. An interval is written with its endpoints in parentheses or brackets. For instance, the inequality \(-1 < y < 15\) translates into the interval notation \((-1, 15)\). Here:\
algebraic manipulation
Algebraic manipulation is a key skill in solving inequalities. This involves performing operations to isolate the variable and solve the inequality step by step. In our example, consider the inequality \(2|7-y|+1<17\). The goal is to isolate the absolute value expression first. By subtracting 1 from both sides, we get \(2|7-y| < 16\). Then, divide both sides by 2, resulting in \(|7-y| < 8\). Now we have two simpler inequalities: \(-8 < 7-y < 8\). To solve these, we do as follows:

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