Problem 68 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(|x-5|=7\)

Step-by-Step Solution

Answer

The solutions to the equation \(|x-5|=7\) are \(x=12\) and \(x=-2\).

Step 1: Set up two separate equations

The first step in solving an absolute value equation is to set up two separate equations. Since \(|x-5|=7\), we make one equation with a positive result and another with a negative. Hence we have \(x-5=7\) and \(x-5=-7\).

Step 2: Solve each equation

Next, solve each equation. For the first one, \(x-5=7\), add 5 to both sides to get \(x=12\). For the second one, \(x-5=-7\), likewise add 5 to both sides to get \(x=-2\)

Step 3: Verify the solutions

It’s always a good idea to check the solutions. In mathematical expressions involving absolute values, not all solutions are valid. In this case plugging 12 and -2 back into the original equation \(|x-5|=7\), we find that both values satisfy the equation, hence they are both valid solutions.