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

Verified step by step guidance

1. Start by understanding that the equation \(|\frac{5}{x-3}| = 10\) involves an absolute value, which means the expression inside can be either positive or negative.
2. Set up two separate equations to account for both scenarios: \(\frac{5}{x-3} = 10\) and \(\frac{5}{x-3} = -10\).
3. For the first equation, \(\frac{5}{x-3} = 10\), multiply both sides by \(x-3\) to eliminate the fraction, resulting in \(5 = 10(x-3)\).
4. Solve the equation \(5 = 10(x-3)\) by distributing the 10 and then isolating \(x\).
5. For the second equation, \(\frac{5}{x-3} = -10\), repeat the process: multiply both sides by \(x-3\) to get \(5 = -10(x-3)\), then solve for \(x\) by distributing and isolating \(x\).

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For example, |x| = 5 means x can be either 5 or -5. In the context of the equation |5/(x-3)| = 10, understanding absolute value is crucial for determining the two possible cases that arise from the equation.

Equations and Solutions

An equation is a mathematical statement that asserts the equality of two expressions. Solving an equation involves finding the values of the variable that make the equation true. In this case, we need to isolate x in the equation |5/(x-3)| = 10, which will lead to two separate equations to solve.

Rational Expressions

A rational expression is a fraction where the numerator and/or denominator are polynomials. In the equation |5/(x-3)| = 10, the expression 5/(x-3) is a rational expression. Understanding how to manipulate and solve rational expressions is essential for finding the values of x that satisfy the equation.