Problem 8 - Algebra and Trigonometry

Question

\(5-22=\) Solve the equation. $$ \frac{1}{2}|x|-7=2 $$

Step-by-Step Solution

Answer

The solutions are x = 18 and x = -18.

Step 1: Isolate the Absolute Value Term

First, add 7 to both sides of the equation to isolate the absolute value term. \[ \frac{1}{2}|x| - 7 + 7 = 2 + 7 \] This simplifies to: \[ \frac{1}{2}|x| = 9 \]

Step 2: Eliminate the Fraction

Multiply both sides of the equation by 2 to eliminate the fraction in front of the absolute value term.\[ 2 \times \frac{1}{2}|x| = 9 \times 2 \]This gives: \[ |x| = 18 \]

Step 3: Solve the Two Possible Equations

Since \(|x| = 18\), we have two possible equations:1. \(x = 18\)2. \(x = -18\)

Step 4: Write the Solution Set

The solutions to the equation are both values that make the original absolute value equation true. Therefore, the solution set is \(x = 18\) or \(x = -18\).

Key Concepts

Isolating the Absolute Value

When solving an equation involving absolute value, your first step is to isolate the absolute value term. This means you want to have the absolute value expression by itself on one side of the equation. In the exercise, the equation given is:\[ \frac{1}{2}|x| - 7 = 2 \] The absolute value expression here is \(|x|\), and it's modified by a fraction and a subtraction operation. To isolate \(|x|\), you'll first need to undo the subtraction. You do this by adding 7 to both sides of the equation:

• \( \frac{1}{2}|x| - 7 + 7 = 2 + 7 \)

This results in:

• \( \frac{1}{2}|x| = 9 \)

Now \(|x|\) is on its own with only the fraction \( \frac{1}{2} \) in front of it. Once the absolute value is isolated, solving becomes simpler.

Eliminating Fractions

Fractions can complicate equations, so eliminating them simplifies your work. After isolating the absolute value in the previous step, we were left with:\[ \frac{1}{2}|x| = 9 \]The fraction \( \frac{1}{2} \) is attached to \(|x|\). To eliminate the fraction, multiply both sides of the equation by the reciprocal of the fraction. The reciprocal of \( \frac{1}{2} \) is 2, so we multiply both sides by 2:

• \( 2 \times \frac{1}{2}|x| = 9 \times 2 \)

This results in:

• \( |x| = 18 \)

Now, the equation is much easier to handle. By clearing out the fraction, you've simplified your equation to a point where you can now solve for \(x\).

Finding the Solution Set

An absolute value equation often leads to two possible solutions. This happens because absolute value represents the distance from zero, which can be achieved from two directions on a number line: positive and negative. In our simplified equation:\[ |x| = 18 \]This means \(x\) could be 18 units away from zero, either to the positive side or the negative side. Thus, we have two possible equations:

• \( x = 18 \)
• \( x = -18 \)

These two solutions reflect both possible cases for our absolute value equation being satisfied. Therefore, we write our solution set as \(x = 18\) or \(x = -18\). Each value in the solution set, when substituted back into the original equation, should satisfy it, proving that both answers are valid. The solution set represents all the values of \(x\) which make the original equation true.