Problem 51 - Algebra 2

Question

Determine whether each statement is sometimes, always, or never true. Explain your reasoning. For all real numbers \(a\) and \(b, a \neq 0,\) the equation \(|a x+b|=0\) will have exactly one solution.

Step-by-Step Solution

Answer

The statement is always true; the equation has exactly one solution, \(x = -\frac{b}{a}\).

Step 1: Understanding the Absolute Value Equation

The equation given is \(|ax + b| = 0\). An absolute value equation \(|X| = 0\) implies that \(X = 0\) because the absolute value of a number is zero only when the number itself is zero.

Step 2: Setting the Expression Inside the Absolute Value to Zero

Set the expression inside the absolute value to zero: \(ax + b = 0\). This transformation is always valid because, for the absolute value to be zero, the expression itself must equal zero.

Step 3: Solving the Linear Equation

To solve \(ax + b = 0\), isolate \(x\):\[ax = -b\]\[x = -\frac{b}{a}\]This transformation allows us to find the value of \(x\) that makes the expression inside the absolute value zero.

Step 4: Analyzing the Number of Solutions

The equation \(x = -\frac{b}{a}\) gives a unique value for \(x\), as long as \(a eq 0\). Therefore, the equation \(|ax + b| = 0\) has exactly one solution, which is \(x = -\frac{b}{a}\).

Key Concepts

Real Numbers

Real numbers form the basis of many mathematical concepts and equations. They are the numbers that can be found on the number line, encompassing both rational and irrational numbers. Real numbers include:

• Natural numbers (e.g., 1, 2, 3)
• Whole numbers (e.g., 0, 1, 2, 3)
• Integers (e.g., -1, 0, 1)
• Rational numbers (e.g., 1/2, 0.75)
• Irrational numbers (e.g., π, √2)

When dealing with problems in algebra, real numbers are often involved, particularly in equations and expressions. In the context of the exercise, two real numbers, "a" and "b," play a crucial role in forming the equation. Understanding the nature of real numbers helps in analyzing the solutions of equations. It is important to remember that real numbers are continuous and unbroken along the number line, meaning they do not skip any values.

Linear Equations

Linear equations are mathematical statements of equality involving variables of the first degree, meaning the variables are raised to the power of one. They are represented in the standard form as:

• ax + b = 0

In this equation, "a" and "b" are constant coefficients, and "x" is the variable we aim to solve for. A linear equation can be visualized as a straight line when graphed on a coordinate plane. In the exercise, the expression inside the absolute value, \(ax + b\), is a linear equation set to zero to find its solution. Linear equations typically have one solution, particularly when the equation's graphical representation intersects the x-axis at a single point. Solving these equations involves isolating the variable "x" by performing inverse operations to both sides, making it a fundamental skill in algebra.

Solution of Equations

To find the solution of equations, particularly when involving absolute values and linear equations, understanding different steps is crucial. The absolute value equation \(|ax + b| = 0\) requires us to focus on the expression itself, given that absolute values cannot be negative.

Steps to Solve:

Firstly, set the expression inside the absolute value to zero, making it \(ax + b = 0\). This simplification is viable because the only number whose absolute value is zero is zero itself. From here, solving the linear equation involves basic algebra. Rearrange the equation to isolate "x" by:

• Subtracting "b" from both sides: \( ax = -b \)
• Dividing by "a" (as long as \(a eq 0\)) to solve for "x": \( x = -\frac{b}{a} \)

This shows that \( x = -\frac{b}{a} \) is the solution. The uniqueness of this solution confirms that the equation \(|ax + b| = 0\) indeed has exactly one solution, given that "a" is not zero. This whole process exemplifies a practical approach to solving equations, where each step leads logically towards finding the correct solution.