Problem 9 - Algebra for College Students

Question

Solve each equation. $$ |x-3|=9 $$

Step-by-Step Solution

Answer

The solutions are \(x = 12\) and \(x = -6\).

Step 1: 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.

Step 2: Set Up the Two Possible Equations

Based on the property of absolute value, create two separate equations: 1) \(x-3 = 9\) 2) \(x-3 = -9\)

Step 3: Solve the First Equation

Solve the equation \(x-3 = 9\): Add 3 to both sides: \(x - 3 + 3 = 9 + 3\)Simplify: \(x = 12\)

Step 4: Solve the Second Equation

Solve the equation \(x-3 = -9\): Add 3 to both sides: \(x - 3 + 3 = -9 + 3\)Simplify: \(x = -6\)

Step 5: Write the Final Solution

Combine the solutions from both equations: The solutions to the equation \(|x-3| = 9\) are \(x = 12\) and \(x = -6\).

Key Concepts

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.

• \(|positive| = positive\)
• \(|negative| = positive\)

Thus, the problem splits into two separate equations to solve both scenarios.

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:

• For \(x-3=9\), add 3 to both sides to yield \(x=12\).
• For \(x-3=-9\), add 3 to both sides to get \(x=-6\).

Summarizing, solving absolute value equations usually involves:

• Splitting the equation into two scenarios (positive and negative).
• Solving each resulting equation separately.

Lastly, combine the solutions from both scenarios.

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:

• Add 3 to both sides of \(x-3=9\) to get \(x=12\).
• Add 3 to both sides of \(x-3=-9\) to get \(x=-6\).

Solving linear equations usually requires simple arithmetic operations like addition, subtraction, multiplication, or division. Mastery of linear equations is foundational for understanding more complex algebraic concepts.

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.

• Understand properties of operations (addition, subtraction)
• Apply absolute value properties
• Solve linear equations

By practicing algebra, students develop strong problem-solving skills and the ability to work with abstract concepts, making it easier to tackle more advanced math topics.