Not isolating the absolute value term

Oct 13, 2025 by leolavanderia.com.br 130 views

Introduction

Absolute value equations can be challenging to solve, but with the right approach, they can be broken down into manageable steps. In this article, we will focus on isolating and rewriting absolute value equations, using a step-by-step approach to solve for the variable x. We will use the equation ${$ \frac{3}{4} + |5-x| = \frac{13}{4}$}$ as an example to illustrate the process.

Step 1: Isolate the Absolute Value

The first step in solving an absolute value equation is to isolate the absolute value term. To do this, we need to subtract ${$ \frac{3}{4}$}$ from both sides of the equation. This will give us:

${$ |5-x| = \frac{13}{4} - \frac{3}{4}$}$

Simplifying the right-hand side of the equation, we get:

${$ |5-x| = \frac{10}{4}$}$

${$ |5-x| = \frac{5}{2}$}$

Step 2: Remove the Absolute Value

Now that we have isolated the absolute value term, we can remove it by considering two cases: one where the expression inside the absolute value is positive, and one where it is negative.

Case 1: Expression Inside Absolute Value is Positive

If the expression inside the absolute value is positive, we can remove the absolute value sign and solve for x. In this case, we have:

${ $5-x = \frac{5}{2}\$}$

Subtracting 5 from both sides of the equation, we get:

${$ -x = -\frac{5}{2}$}$

Multiplying both sides of the equation by -1, we get:

${$ x = \frac{5}{2}$}$

Case 2: Expression Inside Absolute Value is Negative

If the expression inside the absolute value is negative, we can remove the absolute value sign and solve for x. In this case, we have:

${$ -(5-x) = \frac{5}{2}$}$

Distributing the negative sign to the terms inside the parentheses, we get:

${$ -5 + x = \frac{5}{2}$}$

Adding 5 to both sides of the equation, we get:

${$ x = \frac{5}{2} + 5$}$

Simplifying the right-hand side of the equation, we get:

${$ x = \frac{15}{2}$}$

Conclusion

In this article, we have shown how to isolate and rewrite absolute value equations using a step-by-step approach. We have used the equation ${$ \frac{3}{4} + |5-x| = \frac{13}{4}$}$ as an example to illustrate the process. By following the steps outlined in this article, you should be able to solve absolute value equations with confidence.

Common Mistakes to Avoid

When solving absolute value equations, there are several common mistakes to avoid. These include:

Not isolating the absolute value term: Make sure to isolate the absolute value term before removing it.
Not considering both cases: Make sure to consider both cases where the expression inside the absolute value is positive and negative.
Not simplifying the equation: Make sure to simplify the equation after removing the absolute value sign.

Practice Problems

To practice solving absolute value equations, try the following problems:

Solve the equation ${$ |2x-3| = 5$}$.
Solve the equation ${$ |x+2| = 3$}$.
Solve the equation ${$ |2x-1| = 4$}$.

Solutions

${$ |2x-3| = 5$}$

Removing the absolute value sign, we get:

${ $2x-3 = 5\$}$

${ $2x-3 = -5\$}$

Solving for x, we get:

${$ x = 4$}$

${$ x = -1$}$

${$ |x+2| = 3$}$

Removing the absolute value sign, we get:

${$ x+2 = 3$}$

${$ x+2 = -3$}$

Solving for x, we get:

${$ x = 1$}$

${$ x = -5$}$

${$ |2x-1| = 4$}$

Removing the absolute value sign, we get:

${ $2x-1 = 4\$}$

${ $2x-1 = -4\$}$

Solving for x, we get:

${$ x = \frac{5}{2}$}$

${$ x = -\frac{3}{2}$}$

Conclusion

Introduction

In our previous article, we discussed how to isolate and rewrite absolute value equations using a step-by-step approach. In this article, we will answer some of the most frequently asked questions about isolating and rewriting absolute value equations.

Q: What is an absolute value equation?

A: An absolute value equation is an equation that contains an absolute value term. The absolute value of a number is its distance from zero on the number line, without considering direction.

Q: How do I isolate the absolute value term in an equation?

A: To isolate the absolute value term, you need to subtract or add the same value to both sides of the equation. This will give you an equation with the absolute value term on one side and a constant on the other side.

Q: What are the two cases I need to consider when removing the absolute value sign?

A: When removing the absolute value sign, you need to consider two cases:

Case 1: The expression inside the absolute value is positive.
Case 2: The expression inside the absolute value is negative.

Q: How do I know which case to consider first?

A: To determine which case to consider first, you need to look at the expression inside the absolute value. If the expression is positive, you can consider Case 1. If the expression is negative, you can consider Case 2.

Q: What if I get two different solutions for x?

A: If you get two different solutions for x, it means that the equation has two possible values for x. This is a common occurrence when solving absolute value equations.

Q: How do I know which solution to choose?

A: When choosing between two solutions, you need to check if both solutions satisfy the original equation. If both solutions satisfy the equation, you can choose either one. If only one solution satisfies the equation, you can choose that one.

Q: Can I use absolute value equations to solve real-world problems?

A: Yes, absolute value equations can be used to solve real-world problems. For example, you can use absolute value equations to model the distance between two points on a number line.

Q: What are some common mistakes to avoid when solving absolute value equations?

A: Some common mistakes to avoid when solving absolute value equations include:

Not isolating the absolute value term
Not considering both cases
Not simplifying the equation
Not checking if both solutions satisfy the original equation

Q: How can I practice solving absolute value equations?

A: You can practice solving absolute value equations by working through example problems and checking your solutions. You can also try solving absolute value equations with different coefficients and constants.

Q: What are some advanced topics related to absolute value equations?

A: Some advanced topics related to absolute value equations include:

Solving absolute value inequalities
Solving systems of absolute value equations
Using absolute value equations to model real-world problems

Conclusion

In this article, we have answered some of the most frequently asked questions about isolating and rewriting absolute value equations. We hope that this article has been helpful in clarifying any confusion you may have had about absolute value equations. If you have any further questions, please don't hesitate to ask.

Practice Problems

To practice solving absolute value equations, try the following problems:

Solve the equation ${$ |2x-3| = 5$}$.
Solve the equation ${$ |x+2| = 3$}$.
Solve the equation ${$ |2x-1| = 4$}$.

Solutions

${$ |2x-3| = 5$}$

Removing the absolute value sign, we get:

${ $2x-3 = 5\$}$

${ $2x-3 = -5\$}$

Solving for x, we get:

${$ x = 4$}$

${$ x = -1$}$

${$ |x+2| = 3$}$

Removing the absolute value sign, we get:

${$ x+2 = 3$}$

${$ x+2 = -3$}$

Solving for x, we get:

${$ x = 1$}$

${$ x = -5$}$

${$ |2x-1| = 4$}$

Removing the absolute value sign, we get:

${ $2x-1 = 4\$}$

${ $2x-1 = -4\$}$

Solving for x, we get:

${$ x = \frac{5}{2}$}$

${$ x = -\frac{3}{2}$}$