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Magazine Chapter 1: Problem 9
Solve each equation. $$ |x-3|=9 $$
Short Answer
Expert verified
The solutions are \(x = 12\) and \(x = -6\).
Step by step solution
01
Understand the Absolute Value Equation
The equation \(|x-3| = 9\) means that the expression inside the absolute value, \(x-3\), can be either 9 or -9, because the absolute value of both 9 and -9 is 9.
02
Set Up the Two Possible Equations
Based on the property of absolute value, create two separate equations: 1) \(x-3 = 9\) 2) \(x-3 = -9\)
03
Solve the First Equation
Solve the equation \(x-3 = 9\): Add 3 to both sides: \(x - 3 + 3 = 9 + 3\)Simplify: \(x = 12\)
04
Solve the Second Equation
Solve the equation \(x-3 = -9\): Add 3 to both sides: \(x - 3 + 3 = -9 + 3\)Simplify: \(x = -6\)
05
Write the Final Solution
Combine the solutions from both equations: The solutions to the equation \(|x-3| = 9\) are \(x = 12\) and \(x = -6\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value
Absolute value refers to the distance a number is from zero on a number line, regardless of direction. It is always non-negative. For instance, the absolute value of both 9 and -9 is 9. We denote the absolute value of a number 'a' using the symbol \(|a|\). This means \(|9| = 9\) and \(|-9| = 9\).Understanding absolute values is crucial because when solving absolute value equations, we need to consider both the positive and negative scenarios.
In the given exercise, \(|x-3| = 9\), the expression inside the absolute value, \(x-3\), could be either 9 or -9, since the absolute value of both 9 and -9 is 9.
In the given exercise, \(|x-3| = 9\), the expression inside the absolute value, \(x-3\), could be either 9 or -9, since the absolute value of both 9 and -9 is 9.
- \(|positive| = positive\)
- \(|negative| = positive\)
Equation Solving
Equation solving involves finding the value(s) of the variable(s) that satisfy the equation. In our exercise \(|x-3|=9\), we solve by splitting into two equations: \(x-3 = 9\) and \(x-3 = -9\).
Here's how we solve both:
Here's how we solve both:
- For \(x-3=9\), add 3 to both sides to yield \(x=12\).
- For \(x-3=-9\), add 3 to both sides to get \(x=-6\).
- Splitting the equation into two scenarios (positive and negative).
- Solving each resulting equation separately.
Linear Equations
Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. They have the standard form \(ax + b = c\), where a, b, and c are constants.
In our exercise, once we split \(|x-3| = 9\) into simpler equations, we get two linear equations: \(x-3=9\) and \(x-3=-9\).
To solve them:
In our exercise, once we split \(|x-3| = 9\) into simpler equations, we get two linear equations: \(x-3=9\) and \(x-3=-9\).
To solve them:
- Add 3 to both sides of \(x-3=9\) to get \(x=12\).
- Add 3 to both sides of \(x-3=-9\) to get \(x=-6\).
Algebra
Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. It allows us to solve equations and find unknown values.
In the exercise \(|x-3| = 9\), algebra helps us understand and apply the properties of absolute values to set up two possible equations: \(x-3=9\) and \(x-3=-9\). We then use basic algebraic skills to solve these linear equations.
In the exercise \(|x-3| = 9\), algebra helps us understand and apply the properties of absolute values to set up two possible equations: \(x-3=9\) and \(x-3=-9\). We then use basic algebraic skills to solve these linear equations.
- Understand properties of operations (addition, subtraction)
- Apply absolute value properties
- Solve linear equations
