Problem 19 - Algebra for College Students

Question

Solve each equation. $$ \left|12-\frac{1}{2} x\right|=6 $$

Step-by-Step Solution

Answer

The solutions are \(x = 12\) and \(x = 36\).

Step 1 - Understand the Absolute Value Equation

An absolute value equation \(|a| = b\) implies two equations: \(a = b\) and \(a = -b\). Write the given equation in this form.

Step 2 - Set Up Two Equations

Separate the given absolute value equation into two equations: \(12 - \frac{1}{2}x = 6\) and \(12 - \frac{1}{2}x = -6\).

Step 3 - Solve the First Equation

Solve \(12 - \frac{1}{2}x = 6\). Subtract 12 from both sides to get \(-\frac{1}{2}x = -6\). Then, multiply both sides by \-2\ to solve for \x\.

Step 4 - Solve for x

Simplify to get \(x = 12\).

Step 5 - Solve the Second Equation

Solve \(12 - \frac{1}{2}x = -6\). Subtract 12 from both sides to get \(-\frac{1}{2}x = -18\). Then, multiply both sides by \-2\ to solve for \x\.

Step 6 - Solve for x

Simplify to get \(x = 36\).

Step 7 - Verify Solutions

Verify the solutions \(x = 12\) and \(x = 36\) by substituting back into the original equation to ensure that both satisfy \(|12 - \frac{1}{2} x| = 6\).

Key Concepts

headline of the respective core concept

Absolute value is a fundamental concept to understand when solving equations like the one in this exercise. Absolute value represents the distance of a number from 0 on a number line, regardless of direction. It is always non-negative. For any number \(a\), the absolute value is written as \(|a|\). Therefore, solving an absolute value equation like \(|12 - \frac{1}{2} x| = 6\) requires considering both the positive and negative scenarios.

headline of the respective core concept

An algebraic equation is a statement that asserts the equality of two expressions. In our case, the original problem is \(|12 - \frac{1}{2} x| = 6\). To solve this, we need to convert the absolute value equation into two separate linear equations. This is because \(|a| = b\) can be split into two cases: \(a = b\) and \(a = -b\). That means we must solve both \(12 - \frac{1}{2} x = 6\) and \(12 - \frac{1}{2} x = -6\). These steps lead us to find the variable values.

headline of the respective core concept

Solutions verification is essential to ensure the correctness of the solutions obtained. In this exercise, after finding \(x = 12\) and \(x = 36\), we substitute them back into the original equation \(|12 - \frac{1}{2} x| = 6\) to confirm they satisfy the equation. Both values should make the left side equal to 6 when calculated, thus verifying the solutions.