Not considering both cases

Oct 13, 2025 by miodeya.com 130 views

Understanding the Absolute Value Equation

The given equation is $|4x+5|-11=0$ . To solve this equation, we need to isolate the absolute value expression and then consider the two possible cases that arise from the definition of absolute value.

Isolating the Absolute Value Expression

First, we add 11 to both sides of the equation to isolate the absolute value expression:

|4x+5|-11+11=0+11

This simplifies to:

|4x+5|=11

Considering the Two Possible Cases

Since the absolute value of an expression is equal to a positive number, we can consider two possible cases:

Case 1: $4x+5=11$
Case 2: $4x+5=-11$

Solving Case 1

For Case 1, we have:

4x+5=11

Subtracting 5 from both sides gives:

4x=6

Dividing both sides by 4 gives:

x=\frac{6}{4}=\frac{3}{2}=1.5

Solving Case 2

For Case 2, we have:

4x+5=-11

Subtracting 5 from both sides gives:

4x=-16

Dividing both sides by 4 gives:

x=-\frac{16}{4}=-4

Combining the Solutions

We have found two possible solutions for the equation: $x=1.5$ and $x=-4$ . To determine which of these solutions is correct, we need to check if they satisfy the original equation.

Checking the Solutions

Substituting $x=1.5$ into the original equation gives:

|4(1.5)+5|-11=|6+5|-11=|11|-11=0

This shows that $x=1.5$ is a solution to the equation.

Substituting $x=-4$ into the original equation gives:

|4(-4)+5|-11=|-16+5|-11=|-11|-11=0

This shows that $x=-4$ is also a solution to the equation.

Conclusion

We have found that the equation $|4x+5|-11=0$ has two solutions: $x=1.5$ and $x=-4$ . Therefore, the correct answer is:

B. $1.5$ and $-4$

Final Answer

The final answer is $\boxed{B. $1.5$ and $-4$ }$.

Q: What is an absolute value equation?

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

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to isolate the absolute value expression and then consider the two possible cases that arise from the definition of absolute value.

Q: What are the two possible cases for an absolute value equation?

A: The two possible cases for an absolute value equation are:

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

Q: How do I solve Case 1 of an absolute value equation?

A: To solve Case 1, you need to set the expression inside the absolute value equal to the positive number and then solve for the variable.

Q: How do I solve Case 2 of an absolute value equation?

A: To solve Case 2, you need to set the expression inside the absolute value equal to the negative number and then solve for the variable.

Q: What if the absolute value equation has no solution?

A: If the absolute value equation has no solution, it means that the two cases are contradictory, and there is no value of the variable that satisfies the equation.

Q: Can an absolute value equation have more than one solution?

A: Yes, an absolute value equation can have more than one solution. This occurs when both cases have a solution.

Q: How do I check if a solution satisfies the original equation?

A: To check if a solution satisfies the original equation, you need to substitute the solution into the original equation and see if it is true.

Q: What is the importance of absolute value equations in real-life applications?

A: Absolute value equations have many real-life applications, such as:

Modeling distance and time problems
Solving problems involving temperature and cost
Analyzing data and making predictions

Q: Can absolute value equations be used to model real-world problems involving finance?

A: Yes, absolute value equations can be used to model real-world problems involving finance, such as:

Calculating the cost of a loan
Determining the profit or loss of a business
Analyzing the impact of interest rates on investments

Q: How do I use absolute value equations to solve problems involving finance?

A: To use absolute value equations to solve problems involving finance, you need to:

Identify the variables: Identify the variables involved in the problem, such as the cost of a loan or the profit of a business.
Set up the equation: Set up the absolute value equation based on the problem.
Solve the equation: Solve the absolute value equation to find the solution.
Interpret the results: Interpret the results in the context of the problem.

Q: Can absolute value equations be used to model real-world problems involving science and technology?

A: Yes, absolute value equations can be used to model real-world problems involving science and technology, such as:

Modeling the motion of objects
Analyzing the behavior of electrical circuits
Determining the optimal solution to a problem

Q: How do I use absolute value equations to solve problems involving science and technology?

A: To use absolute value equations to solve problems involving science and technology, you need to:

Identify the variables: Identify the variables involved in the problem, such as the position of an object or the voltage of a circuit.
Set up the equation: Set up the absolute value equation based on the problem.
Solve the equation: Solve the absolute value equation to find the solution.
Interpret the results: Interpret the results in the context of the problem.

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 considering both cases: Failing to consider both cases of the absolute value equation can lead to incorrect solutions.
Not checking the solutions: Failing to check the solutions in the original equation can lead to incorrect solutions.
Not interpreting the results correctly: Failing to interpret the results in the context of the problem can lead to incorrect conclusions.

Q: How can I practice solving absolute value equations?

A: To practice solving absolute value equations, you can:

Work on practice problems: Work on practice problems to develop your skills and build your confidence.
Use online resources: Use online resources, such as video tutorials and practice exercises, to help you learn and practice.
Seek help from a teacher or tutor: Seek help from a teacher or tutor if you are struggling with absolute value equations.

Q: What are some real-world applications of absolute value equations?

A: Some real-world applications of absolute value equations include:

Modeling distance and time problems: Absolute value equations can be used to model distance and time problems, such as calculating the time it takes to travel a certain distance.
Solving problems involving temperature and cost: Absolute value equations can be used to solve problems involving temperature and cost, such as calculating the cost of heating or cooling a building.
Analyzing data and making predictions: Absolute value equations can be used to analyze data and make predictions, such as predicting the number of customers a business will have based on past sales data.

Q: Can absolute value equations be used to model real-world problems involving social sciences?

A: Yes, absolute value equations can be used to model real-world problems involving social sciences, such as:

Modeling population growth: Absolute value equations can be used to model population growth, such as calculating the rate of population growth.
Analyzing economic data: Absolute value equations can be used to analyze economic data, such as calculating the rate of inflation.
Determining the optimal solution to a problem: Absolute value equations can be used to determine the optimal solution to a problem, such as determining the optimal price for a product.

Q: How do I use absolute value equations to solve problems involving social sciences?

A: To use absolute value equations to solve problems involving social sciences, you need to:

Identify the variables: Identify the variables involved in the problem, such as the population growth rate or the inflation rate.
Set up the equation: Set up the absolute value equation based on the problem.
Solve the equation: Solve the absolute value equation to find the solution.
Interpret the results: Interpret the results in the context of the problem.

Q: What are some common mistakes to avoid when using absolute value equations to solve problems involving social sciences?

A: Some common mistakes to avoid when using absolute value equations to solve problems involving social sciences include:

Not considering the context of the problem: Failing to consider the context of the problem can lead to incorrect solutions.
Not interpreting the results correctly: Failing to interpret the results in the context of the problem can lead to incorrect conclusions.
Not using the correct variables: Failing to use the correct variables can lead to incorrect solutions.

Q: How can I practice using absolute value equations to solve problems involving social sciences?

A: To practice using absolute value equations to solve problems involving social sciences, you can:

Work on practice problems: Work on practice problems to develop your skills and build your confidence.
Use online resources: Use online resources, such as video tutorials and practice exercises, to help you learn and practice.
Seek help from a teacher or tutor: Seek help from a teacher or tutor if you are struggling with absolute value equations.

Q: What are some real-world applications of absolute value equations in social sciences?

A: Some real-world applications of absolute value equations in social sciences include:

Modeling population growth: Absolute value equations can be used to model population growth, such as calculating the rate of population growth.
Analyzing economic data: Absolute value equations can be used to analyze economic data, such as calculating the rate of inflation.
Determining the optimal solution to a problem: Absolute value equations can be used to determine the optimal solution to a problem, such as determining the optimal price for a product.

Q: Can absolute value equations be used to model real-world problems involving business and economics?

A: Yes, absolute value equations can be used to model real-world problems involving business and economics, such as:

Modeling the cost of a loan: Absolute value equations can be used to model the cost of a loan, such as calculating the interest rate.
Analyzing the behavior of a business: Absolute value equations can be used to analyze the behavior of a business, such as calculating the profit or loss.
Determining the optimal solution to a problem: Absolute value equations can be used to determine the optimal solution to a problem, such as determining the optimal price for a product.

Q: How do I use absolute value equations to solve problems involving business and economics?

A: To use absolute value equations to solve problems involving business and economics, you need to:

Identify the variables: Identify the variables involved in the problem, such as the interest rate or the profit or loss.
Set up the equation: Set up the absolute value equation based on the problem.
Solve the equation: Solve the absolute value equation to find the solution.
Interpret the results: Interpret