Problem 21 - Algebra for College Students

Question

Solve each equation. $$ |0.5 x|=6 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the absolute value equation

The equation \( |0.5x| = 6 \) involves an absolute value. The absolute value of a number represents its distance from 0 on the number line, regardless of its sign. Thus, \( |0.5x| = 6 \) can be split into two separate equations: \( 0.5x = 6 \) and \( 0.5x = -6 \).

Step 2: Solve the first equation

Solve \( 0.5x = 6 \) by isolating \( x \). To do this, multiply both sides of the equation by \( 2 \) to get rid of the coefficient 0.5: \[ 0.5x = 6 \Rightarrow x = 6 \times 2 = 12 \]

Step 3: Solve the second equation

Solve \( 0.5x = -6 \) similarly by isolating \( x \). Multiply both sides of the equation by \( 2 \): \[ 0.5x = -6 \Rightarrow x = -6 \times 2 = -12 \]

Step 4: State the solutions

The solutions to the equation \( |0.5x| = 6 \) are \( x = 12 \) and \( x = -12 \).

Key Concepts

absolute value

Absolute value refers to the distance of a number from zero on the number line. It is always a non-negative number regardless of whether the original number was positive or negative. For instance, the absolute value of both -3 and 3 is 3. In mathematical terms, the absolute value of a number \( x \) is denoted by \( |x| \). For example, \( |3| = 3 \) and \( |-3| = 3 \). Absolute values are crucial when solving equations involving absolute terms because they mean the original number could be either positive or negative.

equation solving

Solving equations is one of the foundations of algebra. You need to find the value of the variable that makes the equation true. The process generally involves isolating the variable by performing inverse operations. For example, when solving \( |0.5x| = 6 \), you split it into two cases due to the nature of the absolute value: \(0.5x = 6\) and \(0.5x = -6\). Each case is then solved separately by isolating \( x \). This method ensures all possible solutions are found.

linear equations

A linear equation is an equation where the highest power of the variable is one. These equations take the form \(ax + b = 0\). The equation from our exercise, \(0.5x=6\), is a linear equation since it can be simplified to \(x = 12\) once the coefficient is dealt with. In general, you solve linear equations by isolating the variable on one side using inverse operations such as addition, subtraction, multiplication, or division.

multiplication property of equality

The multiplication property of equality states that you can multiply both sides of an equation by the same non-zero number without changing the equality. For example, in the step where we go from \(0.5x = 6\) to \(x = 12\), we multiply both sides by 2. This property is fundamental in solving equations because it allows us to manipulate the equation to isolate the variable. Apply this property carefully, ensuring you multiply or divide both sides by the same number.