Problem 23 - Algebra for College Students

Question

Solve each equation. $$ \left|\frac{1}{2} x+3\right|=2 $$

Step-by-Step Solution

Answer

x = -2 and x = -10

Step 1: Understand the absolute value equation

An absolute value equation \(|A| = B\) means that A can be either B or -B. Therefore, the equation \(\begin{vmatrix}\frac{1}{2} x + 3\begin{vmatrix} = 2\) can be split into two separate equations: \(\frac{1}{2} x + 3 = 2\) and \(\frac{1}{2} x + 3 = -2\).

Step 2: Solve the first equation

Solve the equation \(\frac{1}{2} x + 3 = 2 \) for x. First, subtract 3 from both sides to get \(\frac{1}{2} x = -1\). Next, multiply both sides by 2 to isolate x: \( x = -2 \).

Step 3: Solve the second equation

Solve the equation \(\frac{1}{2} x + 3 = -2 \) for x. First, subtract 3 from both sides to get \(\frac{1}{2} x = -5\). Next, multiply both sides by 2 to isolate x: \( x = -10 \).

Step 4: Combine the solutions

The solutions to the equation are the values of x obtained from both equations. Therefore, the solutions are \( x = -2 \) and \( x = -10 \).

Key Concepts

Understanding Absolute Value Equations

Absolute value equations look a bit tricky at first, but they are quite manageable once you know how to handle them. The absolute value of a number is its distance from zero on a number line, regardless of the direction. In simpler terms, it's always positive. For any number A, its absolute value \( \left| A \right| \) will be equal to B if A is either B or \ (-B) \. This gives us two possible equations to solve for in absolute value equations. For example, with \( \left| \frac{1}{2} x + 3 \right| = 2 \), it means we need to solve both \( \frac{1}{2} x + 3 = 2 \) and \ (\frac{1}{2} x + 3 = -2) \. Breaking down absolute value equations into these two cases makes them easier to tackle.

Solving Linear Equations

Linear equations are fundamental in algebra. They usually come in the form of \( Ax + B = C \), where A, B, and C are constants. Our goal is to isolate the variable (commonly x). Let's look at the equations we derived from our absolute value problem: \( \frac{1}{2} x + 3 = 2 \) and \( \frac{1}{2} x + 3 = -2 \. First, we need to simplify both equations step by step:

For \ \frac{1}{2} x + 3 = 2 \, we subtract 3 from both sides to get \ \frac{1}{2} x = -1 \. Then we multiply both sides by 2 to find \ x = -2 \. Similarly, for \ \frac{1}{2} x + 3 = -2 \, we do the same operations: subtract 3 (to get \$ \frac{1}{2} x = -5 \$) and then multiply by 2 (resulting in \ x = -10 \). Linear equations are always solved by performing operations that 'undo' what's being done to the variable.

Algebra for Students

Algebra might seem like a puzzle, but it's all about recognizing patterns and solving for unknowns. When dealing with equations, always remember that what you do to one side, you must do to the other. This keeps everything balanced, just like a scale. Here are some tips to make life easier:

• Identify and understand the type of equation
• Break the problem into smaller, manageable steps
• Perform operations carefully and in the correct order
• Always double-check your solutions

When you see an absolute value equation, remember you're solving two separate linear equations. Breaking down complicated problems into simpler steps not only helps solve the problem but also builds a strong foundation for more advanced math topics. Algebra is a skill that improves with practice, so keep at it, and don't get discouraged!