Solve each equation. | 5/ (x-3) | = 10

Verified step by step guidance

1. Identify the equation given: \(\left| \frac{5}{x} - 3 \right| = 10\). The absolute value expression means the quantity inside the absolute value can be either positive or negative 10.
2. Set up two separate equations to remove the absolute value: 1) \(\frac{5}{x} - 3 = 10\) 2) \(\frac{5}{x} - 3 = -10\)
3. Solve the first equation \(\frac{5}{x} - 3 = 10\) by isolating \(\frac{5}{x}\): add 3 to both sides to get \(\frac{5}{x} = 13\). Then solve for \(x\) by multiplying both sides by \(x\) and dividing both sides by 13.
4. Solve the second equation \(\frac{5}{x} - 3 = -10\) by isolating \(\frac{5}{x}\): add 3 to both sides to get \(\frac{5}{x} = -7\). Then solve for \(x\) by multiplying both sides by \(x\) and dividing both sides by -7.
5. Check the solutions to ensure they do not make the denominator zero and verify if they satisfy the original absolute value equation.

Key Concepts

Absolute Value Equations

An absolute value equation involves expressions within absolute value bars, which represent the distance from zero on the number line. To solve |A| = B, where B is positive, split the equation into two cases: A = B and A = -B, then solve each separately.

Isolating the Variable

Before solving an equation, isolate the variable term on one side to simplify the problem. This often involves performing inverse operations such as addition, subtraction, multiplication, or division to both sides of the equation.

Checking for Extraneous Solutions

When solving absolute value equations, some solutions may not satisfy the original equation due to the nature of absolute values. Always substitute solutions back into the original equation to verify their validity.