Final answer:
Solving an absolute value equation typically results in two scenarios considering both the positive and negative outcomes of the absolute value, whereas solving a linear equation generally has a single solution. Both require algebraic manipulation and checking the reasonableness of the solution.
Step-by-step explanation:
When solving an absolute value equation, you are looking for all the possible solutions that could result in the absolute value expression to match the given number. Because the absolute value of a number is always non-negative, you must consider both the positive and negative scenarios that would result in that value when you remove the absolute value symbol. On the other hand, a linear equation typically has a single solution and you solve for the variable by isolating it on one side of the equation.
The processes are similar because both types of equations require you to perform algebraic manipulations such as adding, subtracting, multiplying or dividing to solve for the variable in question. However, for an absolute value equation, this manipulation leads to two different equations that stem from the fact that the inside of the absolute value can be either positive or negative.
To illustrate, consider the absolute value equation |x-3|=5. This equation has two possible scenarios:
- x - 3 = 5
- x - 3 = -5
Solving these equations yields x = 8 and x = -2, respectively. Comparatively, for a linear equation such as y = 2x + 3, there is only one solution for y when you substitute a specific value for x.
For both types of equations, it is essential to check your answer to ensure it makes sense and applies to the original equation. This step confirms the reasonableness of your result and is a commonality in the problem-solving process for all mathematical equations.