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Chapter 4: Problem 37

Solve. Write the answer using set notation. $$ |5-m|+9=16 $$

Short Answer

Expert verified
{-2, 12}

Step by step solution

01

- Isolate the Absolute Value

Start by isolating the absolute value expression on one side of the equation. Subtract 9 from both sides:\[|5 - m| + 9 = 16\]\[|5 - m| = 16 - 9\]\[|5 - m| = 7\]
02

- Set Up Two Equations

Since the absolute value of a number is always positive, set up two equations to account for both the positive and negative scenarios:\[5 - m = 7\]\[5 - m = -7\]
03

- Solve the First Equation

Solve for \(m\) in the first equation:\[5 - m = 7\]Subtract 5 from both sides:\[-m = 2\]Multiply by -1:\[m = -2\]
04

- Solve the Second Equation

Solve for \(m\) in the second equation:\[5 - m = -7\]Subtract 5 from both sides:\[-m = -12\]Multiply by -1:\[m = 12\]
05

- Write the Solution Set

Combine both solutions to write the answer in set notation:\[\{m | m = -2 \text{ or } m = 12\}\]

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Key Concepts

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

Solving Absolute Value Equations
To solve absolute value equations, it's crucial to understand that absolute values measure the distance from zero. Hence, they always result in positive numbers or zero. Begin by setting up two separate equations to account for the positive and negative scenarios of the value inside the absolute value.

For example, if you're solving \(|5 - m| + 9 = 16\), follow these steps:
  • Isolate the absolute value expression by moving any constants outside the absolute value to the opposite side of the equation.
  • Set up two separate equations. If the equation inside the absolute value is \(|A| = B\), then \((A = B)\) and \((A = -B)\).
  • Solve each equation separately to find the possible values.
This method helps you understand that the equation represents two potential situations since any absolute value expression can have two results.
Set Notation
Set notation is a mathematical method to represent a set of numbers, often used for solutions to equations. It compactly lists all possible solutions to an equation.

Here are key elements when using set notation:
  • Use curly braces \({ ... }\) to enclose the set.
  • Inside the braces, list elements that belong to the set. For multiple elements, use a comma to separate them.
For our example, the solution to \(|5 - m| + 9 = 16\) was \((m = -2)\) and \((m = 12)\). In set notation, we write these solutions as:
\[ \{m | m = -2 \text{ or } m = 12\} \] This indicates that \((m)\) can be either \((-2)\) or \((12)\).
Isolating Absolute Value Expressions
To solve an absolute value equation, the first step usually involves isolating the absolute value expression on one side of the equation. This simplifies the process and makes it easier to consider both positive and negative cases.

Steps to isolate an absolute value expression:
  • Identify the absolute value term in the equation.
  • Move any constants or other terms to the opposite side of the equation by performing basic arithmetic operations.
For instance, starting with \[ |5 - m| + 9 = 16 \] you subtract \((9)\) from both sides to get:
\[|5 - m| = 16 - 9 \]
\[|5 - m| = 7 \] Now, the absolute value expression \[ |5 - m| \] is isolated. This prepares you to handle the equation by addressing the positive and negative scenarios, ultimately leading to the correct solutions.

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Most popular questions from this chapter

Write an equivalent inequality using absolute value. \(x\) is less than 1 unit from 5

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