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Chapter 1: Problem 86

Solve the inequality and write the solution set in interval notation. \(2<|y|<11\)

Short Answer

Expert verified
The solution set in interval notation is: (-11, -2) 鈭� (2, 11).

Step by step solution

01

Understand the absolute value inequality

The inequality involves an absolute value, which means we need to consider both the positive and negative cases of the expression inside the absolute value.
02

Break down the inequality

The given inequality is: 2 < |y| < 11 This means that the absolute value of y must be greater than 2 and less than 11.
03

Split the inequality into two cases

The absolute value inequality 2 < |y| < 11 can be split into two separate inequalities: -11 < y < -2 and 2 < y < 11.
04

Combine the solution sets

The solution set is the combination of both inequalities: -11 < y < -2 or 2 < y < 11.
05

Write the solution in interval notation

The intervals corresponding to the solutions are: (-11, -2) and (2, 11) Therefore, in interval notation, the solution set is: (-11, -2) 鈭� (2, 11)

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Key Concepts

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

absolute value
Absolute value describes the distance of a number from zero on a number line, without considering which direction the number lies. It's always a non-negative value. For example, the absolute value of both -5 and 5 is 5.

Absolute value is indicated by two vertical bars, like this: \(|y|\). When solving inequalities involving absolute values, consider both the positive and negative cases of the expression inside the absolute value. This is because the distance from zero can be achieved by moving in both directions on the number line.
inequality
Inequalities are mathematical expressions that describe the relationship between two values, indicating if one is greater, less, greater than or equal to, or less than or equal to the other. In our example, we need to solve the inequality containing an absolute value: \ 2 < |y| < 11 \. This tells us that the absolute value of \(y\) must be greater than 2 and less than 11.

When dealing with inequalities, always account for the positive and negative solutions. This is important because absolute value inequalities involve two scenarios: \( y \textgreater -2 \textgreater -11 \) and \( 2 \textless y \textless 11 \). Hence, the solution is derived by combining both cases.
interval notation
Interval notation is a way of describing a set of numbers between two endpoints. The notation uses round brackets \(( )\) to signify that an endpoint is not included in the interval and square brackets \(( [ ] )\) when an endpoint is included.

For the given inequality \(2 < |y| < 11\), we split it into two parts: \(-11 < y < -2\) and \(2 < y < 11\). These intervals mean y can be anywhere between -11 and -2 (excluding -11 and -2), or between 2 and 11 (excluding 2 and 11).

In interval notation, we write this as \((-11, -2) \cup (2, 11)\). The union sign \(( \cup )\) indicates that y can lie in either of the two intervals.

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