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

Verified step by step guidance

1. Recognize that the equation involves an absolute value expression: \(\left| \frac{5}{x} - 3 \right| = 10\). Recall that \(|A| = B\) means \(A = B\) or \(A = -B\).
2. Set up two separate equations based on the definition of 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: \(\frac{5}{x} = 13\) Then multiply both sides by \(x\) and solve for \(x\).
4. Solve the second equation \(\frac{5}{x} - 3 = -10\) by isolating \(\frac{5}{x}\): Add 3 to both sides: \(\frac{5}{x} = -7\) Then multiply both sides by \(x\) and solve for \(x\).
5. Check for any restrictions on \(x\) (such as \(x \neq 0\)) and verify your solutions by substituting them back into the original equation to ensure they satisfy the 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. This approach accounts for both positive and negative values that satisfy the equation.

Isolating the Variable

Before solving, isolate the variable expression inside the absolute value. This often involves algebraic manipulation such as adding, subtracting, multiplying, or dividing both sides of the equation. Proper isolation ensures the absolute value expression is clearly defined for applying the two-case method.

Solving Rational Equations

When the variable is in the denominator, as in a rational expression, solving requires careful handling to avoid division by zero. After splitting into two cases, multiply both sides by the denominator to clear fractions, then solve the resulting linear equations. Always check for extraneous solutions that make the denominator zero.