Absolute value equations with fractions can seem daunting at first, but with a systematic approach, they can be easily solved. By following a few simple steps, you can solve these equations with confidence.
The Steps to Solve Absolute Value Equations with Fractions
To solve absolute value equations with fractions, follow these steps:
Step 1: Identify the absolute value expression within the equation.
Step 2: Set up two separate equations, one keeping the expression inside the absolute value symbol positive, and the other making it negative.
Step 3: Solve each equation separately.
Step 4: Check if the solutions obtained in Step 3 satisfy the given equation.
Step 5: Write the final solution as a set of solutions.
Let’s take a closer look at these steps with an example.
Example:
Solve the absolute value equation: |2x – 1/3| = 4/3.
Step 1: Identify the absolute value expression within the equation.
In this equation, the absolute value expression is 2x – 1/3.
Step 2: Set up two separate equations.
Equation 1: 2x – 1/3 = 4/3
Equation 2: 2x – 1/3 = -4/3
Step 3: Solve each equation separately.
For Equation 1:
2x – 1/3 = 4/3
2x = 5/3
x = 5/6
For Equation 2:
2x – 1/3 = -4/3
2x = -1/3
x = -1/6
Step 4: Check if the solutions obtained in Step 3 satisfy the given equation.
Checking x = 5/6:
|2(5/6) – 1/3| = |10/6 – 2/6| = |8/6| = 4/3
The left side of the equation equals the right side, so x = 5/6 is a valid solution.
Checking x = -1/6:
|2(-1/6) – 1/3| = |-2/6 – 2/6| = |-4/6| = 4/6 ≠ 4/3
The left side of the equation does not equal the right side, so x = -1/6 is not a valid solution.
Step 5: Write the final solution as a set of solutions.
The solution to the absolute value equation is x = 5/6.
Frequently Asked Questions
Q1: Can I have a negative number inside the absolute value symbol?
Yes, you can have a negative number inside the absolute value symbol.
Q2: When setting up two separate equations, why do we make one equation positive and the other negative?
By creating two equations, we account for both the positive and negative values that satisfy the absolute value equation.
Q3: What happens if there is no absolute value symbol in the equation?
If there is no absolute value symbol in the equation, it is not an absolute value equation and different methods need to be used to solve it.
Q4: Can an absolute value equation have multiple solutions?
Yes, an absolute value equation can have one or more solutions, depending on the given equation.
Q5: What if the absolute value expression cannot be isolated?
If the absolute value expression cannot be isolated, you may need to rearrange the equation by applying algebraic techniques before proceeding with the solving steps.
Q6: How do you check if a solution is valid?
To check if a solution is valid, substitute the value obtained for x back into the original equation and verify if both sides are equal.
Q7: What if none of the solutions satisfy the equation after checking?
If none of the solutions satisfy the equation after checking, there may be no solution to the given absolute value equation.
Q8: Are there other methods to solve absolute value equations with fractions?
Yes, there are other methods, such as graphing or numerical approximation, which can be used to solve absolute value equations with fractions.
Q9: Can I eliminate the absolute value symbol without setting up two separate equations?
No, to solve absolute value equations with fractions, you must set up two separate equations, one positive and one negative, to consider both possible solutions.
Q10: Can the absolute value expression be a fraction?
Yes, the absolute value expression can be a fraction, as long as it follows the necessary steps for solving absolute value equations.
Q11: What if the absolute value expression is on both sides of the equation?
If the absolute value expression is on both sides of the equation, you will need to apply appropriate algebraic techniques to isolate it on one side before proceeding with the solving steps.
Q12: Are absolute value equations with fractions commonly used in real-world scenarios?
Yes, absolute value equations with fractions are often encountered in various real-world applications, such as physics, engineering, and financial modeling. They are valuable tools for solving problems involving distances, rates, and percentages.