Problem 6 - College Algebra with Modeling and Visualization

Question

Exercises \(1-8:\) Let \(a \neq 0\) $$ \text { Solve }|a x+b|=0 $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{b}{a}\).

Step 1: Understand the Absolute Value Equation

The equation \(|ax + b| = 0\) involves an absolute value. An absolute value equation equals zero when the expression inside the absolute value is zero itself.

Step 2: Set the Inside Expression to Zero

Since \(|ax + b| = 0\) implies that \(ax + b = 0\), we can remove the absolute value bars and solve this equation for \(x\).

Step 3: Solve the Linear Equation

The equation to solve is \(ax + b = 0\). Subtract \(b\) from both sides to get \(ax = -b\).

Step 4: Isolate \(x\)

Divide both sides by \(a\), which is not zero, to isolate \(x\). Thus, \(x = -\frac{b}{a}\).

Key Concepts

Linear Equations

Linear equations are one of the foundational elements of algebra. They are characterized by expressions that form straight lines when graphed. A linear equation is any equation that can be written in the form:

• ax + b = c

Where a, b, and c are constants, and x is the variable. These equations are termed "linear" because they graphically represent a line. Among the simplest linear equations is ax + b = 0, as seen in the problem statement.
This equation specifically represents a horizontal line crossing the x-axis at the solution, showing where the expression equals zero. Solving linear equations involves basic algebraic manipulation to isolate the variable x, which will be addressed next.

Solving Equations

The process of solving equations is essentially finding the value of the variable that makes the equation true. When dealing with an absolute value equation like

• e.g., |ax + b| = 0,

we first interpret what it means. The absolute value of a number represents its distance from zero on the number line, thus being zero implies the number itself is zero. Therefore,

• ax + b = 0

extracts removing the absolute value.
To solve

• ax + b = 0,

we perform algebraic operations to isolate the variable x:

• Subtract b from both sides to get ax = -b.
• Divide by a (noting a ≠ 0) to find x = -\( \frac{b}{a} \).

Algebraic Expressions

Understanding algebraic expressions is crucial when working with equations, as they comprise variables and constants combined using operations like addition, subtraction, multiplication, and division. In our problem, the expression

• ax + b

is central.
Here,

• ax represents a term with a variable; the product of the constant coefficient a and the variable x,
• while b is a constant.

In manipulating algebraic expressions, the goal often entails simplifying the expression or solving for the variable. Understanding how coefficients and constants interact allows us to rearrange terms effectively, such as moving b across the equation equal to zero when isolating x.
Mastering basic algebraic expressions enables us to tackle more complex equations by systematically applying similar steps. Properly handling expressions is key to algebraic proficiency.