Problem 61 - College Algebra Essentials

Question

Solve each absolute value equation or indicate that the equation has no solution. $$ |x|=8 $$

Step-by-Step Solution

Answer

The solutions for the equation \(|x|=8\) are \(x = 8\) and \(x = -8\).

Step 1: Understand the equation

The equation given is \(|x|=8\). In this equation, \(x\) is the variable and the absolute value of \(x\) equals 8.

Step 2: Solve for positive value

The first solution is when \(x\) is positive. This occurs when the stuff inside the absolute value equals 8. Therefore, \(x = 8\).

Step 3: Solve for negative value

The second solution is when \(x\) is negative. This occurs when the stuff inside the absolute value equals -8. Therefore, \(x = -8\).

Key Concepts

Solving Equations

When solving equations, especially those involving absolute values, the primary goal is to isolate the variable. The absolute value equation \(|x| = 8\) involves finding all possible values of \(x\) that satisfy the condition. When an equation includes absolute values, you must consider both positive and negative solutions due to the nature of absolute values, which represent distance from zero. In this example, you explore two scenarios: \(x = 8\) and \(x = -8\) to ensure that you comprehensively solve the equation.
The steps are typically:

• Identify the absolute value equation.
• Split into two separate equations based on positive and negative scenarios.
• Solve each equation individually.

By following these steps, you can successfully determine whether an absolute value equation has one, two, or no solutions.

Absolute Values

Absolute values are a fundamental concept in mathematics that measure the distance of a number from zero on the number line. This concept is symbolized by two vertical lines around a number or expression, such as \(|x|\).
For example, both \(|8|\) and \(|-8|\) equal 8 because both numbers are eight units away from zero. When dealing with absolute value equations, it's crucial to understand this property:

• If \(|a| = b\) (where \(b\) is a positive number), then \(a = b\) or \(a = -b\).
• Absolute value equations can have two solutions as they consider both direction and magnitude.
• If \(b = 0\), then \(a\) must also be zero.

Grasping these concepts helps in analyzing and solving equations involving absolute values.

Algebraic Expressions

Algebraic expressions involve variables, constants, and operations such as addition and multiplication. In the context of absolute value equations, understanding algebraic expressions is key to isolating and solving for the variable.
Consider the absolute value equation \(|x| = 8\):

• The expression \(|x|\) denotes the absolute value operation applied to the variable \(x\).
• To solve, express the equation without absolute values by considering both \(x = 8\) and \(x = -8\). This way, the expression transforms into simpler algebraic forms.
• Solving these simple equations involves recognizing algebraic safeguards such as the properties of equality and balancing equations through inverse operations.

By mastering the manipulation of algebraic expressions, solving equations involving absolute values becomes a more straightforward process.