Solve each equation. | x - 4/ 2| = 5

Verified step by step guidance

1. Recognize that the equation involves an absolute value: \(| \frac{x - 4}{2} | = 5\). Recall that \(|A| = B\) means \(A = B\) or \(A = -B\).
2. Set up two separate equations based on the definition of absolute value: \(\frac{x - 4}{2} = 5\) and \(\frac{x - 4}{2} = -5\).
3. Solve the first equation \(\frac{x - 4}{2} = 5\) by multiplying both sides by 2 to eliminate the denominator: \(x - 4 = 10\).
4. Add 4 to both sides of the first equation to isolate \(x\): \(x = 10 + 4\).
5. Solve the second equation \(\frac{x - 4}{2} = -5\) similarly by multiplying both sides by 2: \(x - 4 = -10\), then add 4 to both sides to find \(x = -10 + 4\).

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For an expression |A| = B, where B ≥ 0, the solutions are A = B or A = -B. Understanding this helps in splitting the equation into two separate cases to solve.

Solving Linear Equations

Solving linear equations involves isolating the variable by performing inverse operations such as addition, subtraction, multiplication, or division. After splitting the absolute value equation into two linear equations, each can be solved using these techniques to find the values of x.

Handling Fractions in Equations

When variables appear in fractional form, it is important to clear denominators by multiplying both sides of the equation by the denominator. This simplifies the equation and makes it easier to isolate the variable and solve the equation accurately.