How do you solve absolute value equations with fractions?

How do you solve absolute value equations with fractions?

Solving absolute value equations with fractions requires understanding the properties of absolute value and applying them to the given equation. Here is a step-by-step guide on how to solve such equations:

Step 1: Set up the equation

Begin by writing the given absolute value equation. For example, consider the equation |(x/2) – 3| = 4/5.

Step 2: Isolate the absolute value expression

To isolate the absolute value expression, we need to get rid of the absolute value bars. We do this by separating the equation into two separate equations: one with the positive of the expression inside the absolute value and the other with the negative of the expression.

In the above example,
(x/2) – 3 = 4/5 and (x/2) – 3 = -4/5

Step 3: Solve each equation separately

Solve each equation separately to find the possible values of the variable.

For the equation (x/2) – 3 = 4/5,

Add 3 to both sides to isolate the variable:
(x/2) = 4/5 + 3

Now, find a common denominator (in this case, it is 5) and add the fractions:
(x/2) = 4/5 + 15/5

Simplify the right side of the equation:
(x/2) = 19/5

Multiply both sides of the equation by 2 to solve for x:
x = 19/5 * 2

Simplify:
x = 38/5

Similarly, for the equation (x/2) – 3 = -4/5,

Add 3 to both sides:
(x/2) = -4/5 + 3

Find a common denominator and add the fractions:
(x/2) = -4/5 + 15/5

Simplify:
(x/2) = 11/5

Multiply both sides by 2 to solve for x:
x = 11/5 * 2

Simplify:
x = 22/5

Therefore, the solutions to the absolute value equation |(x/2) – 3| = 4/5 are x = 38/5 and x = 22/5.

FAQs:

1. Can absolute value be negative in an equation?

No, absolute value represents the distance between a number and zero on a number line, and it is always positive or zero.

2. What happens when an absolute value equation has no solution?

If solving an absolute value equation results in a statement that is never true, then the equation has no solution.

3. Is it necessary to isolate the absolute value expression?

Yes, isolating the absolute value expression allows you to deal with two separate equations and makes the solving process easier.

4. How do you solve absolute value equations using inequalities?

To solve an absolute value inequality, set up two separate inequalities, one without an absolute value and the other with the negation of the absolute value, and solve them separately.

5. Can there be multiple solutions for an absolute value equation?

Yes, an absolute value equation can have one, two, or no solution, depending on the specific equation.

6. How do you determine the inequality sign when solving absolute value equations?

The inequality sign depends on whether the absolute value expression is less than or greater than a certain value or equal to it. The appropriate inequality sign must be chosen according to the given equation.

7. Can fractions be part of the absolute value expression?

Yes, fractions can be present in the absolute value expression, and the same principles of solving absolute value equations apply.

8. What is the geometric interpretation of absolute value?

Geometrically, absolute value can be understood as the distance between a number and zero on a number line.

9. Can absolute value be represented by an algebraic expression?

Yes, absolute value can be represented and manipulated algebraically using conditional statements or piecewise functions.

10. Are there any alternative methods to solve absolute value equations?

Several alternative methods, such as graphing or using logical reasoning, can be used to solve absolute value equations depending on the specific problem.

11. Can complex numbers be part of an absolute value equation?

Yes, an absolute value equation can involve complex numbers, although the solutions will be complex as well.

12. Do absolute value equations have real number solutions only?

No, absolute value equations can have real number solutions as well as complex number solutions, depending on the given equation and the value of the absolute value expression.

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