Problem 17 - Calculus Single Variable

Question

Write the set using interval notation. Use the symbol \(\cup\) where appropriate. \(\\{y:|y+4| \leq 10\\}\)

Step-by-Step Solution

Answer

The interval notation is \([-14, 6]\).

Step 1: Understand the Absolute Value Inequality

The expression \(|y+4| \leq 10\) indicates that the distance between \(y+4\) and 0 is less than or equal to 10. This can be split into two inequalities: \(y+4 \leq 10\) and \(y+4 \geq -10\).

Step 2: Solve Each Inequality

Start by solving \(y+4 \leq 10\): Subtract 4 from both sides to get \(y \leq 6\). Similarly, solve \(y+4 \geq -10\): Subtract 4 from both sides to get \(y \geq -14\).

Step 3: Combine the Solutions

The solution to the inequality \(|y+4| \leq 10\) includes all \(y\) values that satisfy both \(y \leq 6\) and \(y \geq -14\). Together, these inequalities form the interval \([-14, 6]\).

Step 4: Write in Interval Notation

The set of all values that satisfy the original inequality is given in interval notation as \([-14, 6]\).

Key Concepts

Understanding Absolute Value Inequalities

Absolute value inequalities involve expressions that measure the distance of a number from zero on the number line. If you encounter an inequality like \(|y+4| \leq 10\), it means the distance from \(y+4\) to zero must be less than or equal to 10. This can be seen as two distinct conditions:

• \(y+4 \leq 10\) (the larger distance is within 10 units)
• \(y+4 \geq -10\) (the smaller distance is within 10 units)

Splitting the absolute value inequality into two separate inequalities helps us see which values satisfy the condition. Remember, absolute value reflects on just how far, rather than which direction (positive or negative), from zero a number can be.

Step-by-Step of Solving Inequalities

To tackle inequalities such as these, resolving the absolute value means you have two clear linear inequalities to solve. Let’s consider the same example:\(|y+4| \leq 10\).
First, you separate it into \(y+4 \leq 10\) and \(y+4 \geq -10\). Solving these involves basic arithmetic:

• For \(y+4 \leq 10\): Subtract 4 from each side, giving \(y \leq 6\).
• For \(y+4 \geq -10\): Similarly, subtract 4 from each side, arriving at \(y \geq -14\).

Once solved separately, as both inequalities need to be true simultaneously, you combine them. The solution for \(|y+4| \leq 10\) thus becomes the set of \(y\) that lie between -14 and 6, inclusive.

Using Set Notation and Interval Notation

Set notation and interval notation are ways to represent a group of numbers that satisfy a given condition. For example, set notation expresses the solution as \(\{ y : -14 \leq y \leq 6 \} \).
Interval notation, on the other hand, uses brackets to define the range of acceptable values. Here, \([-14, 6]\) denotes all numbers \(y\) such that \(-14 \leq y \leq 6\).

• A closed interval uses brackets \([ ]\) to indicate inclusive boundaries.
• An open interval would use parentheses \(( )\) to exclude endpoints.

For the absolute value inequality \(|y+4| \leq 10\), using interval notation simplifies understanding by precisely showing the range of solutions without needing wordy explanations.