Problem 3 - Algebra for College Students

Question

How many solutions will \(|a x+b|=k\) have for each situation? (a) \(k=0\) (b) \(k>0\) (c) \(k<0\)

Step-by-Step Solution

Answer

(a) 1 solution. (b) 2 solutions. (c) No solution.

Step 1: Understanding the Abs Value Equation

The given equation is \(|a x + b| = k\). This is an absolute value equation, meaning it expresses the distance of \(a x + b\) from zero on a number line. To solve these equations, we must consider the properties of absolute values: they are always non-negative.

Step 2: Case (a) \(k=0\)

When \(k = 0\), the equation \(|a x + b| = 0\) implies that \a x + b\ must also equal zero, since the distance from zero is zero only at zero. Thus, \(a x + b = 0\). Solving for \(x\), we get \(x = -\frac{b}{a}\). Hence, there is exactly one solution in this case.

Step 3: Case (b) \(k>0\)

When \(k > 0\), the absolute value equation \(|a x + b| = k\) can be split into two linear equations: \(a x + b = k\) and \(a x + b = -k\). Solving these two equations: 1. For \(a x + b = k\), we find \(x = \frac{k - b}{a}\). 2. For \(a x + b = -k\), we find \(x = \frac{-k - b}{a}\). Thus, there are exactly two solutions when \(k > 0\).

Step 4: Case (c) \(k<0\)

When \(k < 0\), the equation \(|a x + b| = k\) has no solutions because absolute values cannot be negative. Therefore, there are no solutions in this situation.

Key Concepts

Solving Absolute Value Equations

To solve absolute value equations like \(|a x + b| = k\), understanding the nature of absolute value is crucial. An absolute value equation describes the distance of the expression inside the absolute value from zero on a number line.

• When solving equations of the form \(|a x + b| = k\), there are specific steps to take depending on the value of \(k\).
• Case (a): \(k = 0\): When \(k = 0\), the absolute value equation simplifies to \(a x + b = 0\). Solving this, we find \(x = -\frac{b}{a}\), meaning there is only one solution.
• Case (b): \(k > 0\): When \(k\) is positive, we split the equation into two linear equations: \(a x + b = k\) and \(a x + b = -k\). Solving each equation separately gives us two solutions.
• Case (c): \(k < 0\): If \(k\) is negative, there are no solutions. This is because absolute values cannot be negative.

Breaking it down step-by-step helps grasp how to tackle each scenario. Always remember: absolute values denote non-negative distances from zero.

Properties of Absolute Values

Absolute values express the distance from zero, and they are always non-negative. An equation of the form \(|a x + b| = k\) leverages these properties to define specific conditions.

• The absolute value of a number \(x\) is \(x\) if \(x \geq 0\) and \(-x\) if \(x < 0\).
• For example, \(|-3| = 3\) and \( |5| = 5\).
• Absolute value equations, therefore, must be analyzed to ensure the value inside is within the allowed range (non-negative).
• Through these properties, we solve \(|a x + b| = k\) by considering all possible values that the expression \(|a x + b|\) can take.

It's important to note these properties when solving absolute value equations, as they determine the possible solutions depending on whether \(k\) is zero, positive, or negative.

Linear Equations in Absolute Value Problems

Solving absolute value equations often leads to solving linear equations. Linear equations are equations of the form \(a x + b = c\), where \(a\), \(b\), and \(c\) are constants.

• When \(k = 0\), solving \(|a x + b| = 0\) results in the linear equation \(a x + b = 0\), with one unique solution: \(x = -\frac{b}{a}\).
• When \(k > 0\), the equation \(|a x + b| = k\) splits into two linear equations: \(a x + b = k\) and \(a x + b = -k\). Solving these equations provides two distinct solutions.
• Each linear equation can be solved by isolating \(x\) and simplifying the equation to find the value of \(x\).

Understanding how to handle linear equations is essential when working with absolute value problems. Solving them step-by-step ensures accuracy and clarity.