Equations And Inequalities And Absolute ValueSolve ∣ 4 X − 6 ∣ = − 10 |4x - 6| = -10 ∣4 X − 6∣ = − 10 .A. X = − 1 X = -1 X = − 1 Or X = 4 X = 4 X = 4 B. There Are No Solutions.C. X = − 4 X = -4 X = − 4 Or X = 1 X = 1 X = 1 D. X = 4 X = 4 X = 4 Or X = 1 X = 1 X = 1

Mar 12, 2025 by ADMIN 265 views

Equations and Inequalities: Solving Absolute Value Equations

Understanding Absolute Value Equations

Absolute value equations are a type of algebraic equation that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction. In other words, the absolute value of a number is always non-negative. When solving absolute value equations, we need to consider both the positive and negative cases of the expression inside the absolute value bars.

Solving Absolute Value Equations

To solve an absolute value equation, we need to isolate the expression inside the absolute value bars and then consider two cases: one where the expression is positive and one where the expression is negative. Let's consider the equation $|4x - 6| = -10$ .

Step 1: Isolate the Expression Inside the Absolute Value Bars

The first step in solving the equation is to isolate the expression inside the absolute value bars. In this case, the expression is $4x - 6$ . We can rewrite the equation as $4x - 6 = -10$ .

Step 2: Consider the Positive Case

When the expression inside the absolute value bars is positive, we can simply remove the absolute value bars and solve for $x$ . In this case, we have $4x - 6 = -10$ . To solve for $x$ , we can add 6 to both sides of the equation, which gives us $4x = -4$ . Then, we can divide both sides of the equation by 4, which gives us $x = -1$ .

Step 3: Consider the Negative Case

When the expression inside the absolute value bars is negative, we can remove the absolute value bars and change the sign of the expression. In this case, we have $-(4x - 6) = -10$ . To solve for $x$ , we can distribute the negative sign to the expression inside the parentheses, which gives us $-4x + 6 = -10$ . Then, we can subtract 6 from both sides of the equation, which gives us $-4x = -16$ . Finally, we can divide both sides of the equation by -4, which gives us $x = 4$ .

Step 4: Check the Solutions

Now that we have found two possible solutions, $x = -1$ and $x = 4$ , we need to check if they are valid. To do this, we can plug each solution back into the original equation and see if it is true. If the solution is true, then it is a valid solution. If the solution is not true, then it is not a valid solution.

Step 5: Check the Solution $x = -1$

To check the solution $x = -1$ , we can plug it back into the original equation: $|4(-1) - 6| = |-4 - 6| = |-10| = 10 \neq -10$ . Since this is not true, $x = -1$ is not a valid solution.

Step 6: Check the Solution $x = 4$

To check the solution $x = 4$ , we can plug it back into the original equation: $|4(4) - 6| = |16 - 6| = |10| = 10 \neq -10$ . Since this is not true, $x = 4$ is not a valid solution.

Step 7: Conclusion

Since neither of the solutions $x = -1$ and $x = 4$ are valid, we can conclude that there are no solutions to the equation $|4x - 6| = -10$ .

The Final Answer

The final answer is B. There are no solutions.
Equations and Inequalities: Solving Absolute Value Equations - Q&A

Understanding Absolute Value Equations

Frequently Asked Questions

Q: What is an absolute value equation?

A: An absolute value equation is a type of algebraic equation that involves the absolute value of a variable or expression.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to isolate the expression inside the absolute value bars and then consider two cases: one where the expression is positive and one where the expression is negative.

Q: What is the difference between a positive and negative case in an absolute value equation?

A: In a positive case, the expression inside the absolute value bars is positive, and you can simply remove the absolute value bars and solve for x. In a negative case, the expression inside the absolute value bars is negative, and you need to remove the absolute value bars and change the sign of the expression.

Q: How do I check if a solution is valid?

A: To check if a solution is valid, you need to plug it back into the original equation and see if it is true. If the solution is true, then it is a valid solution. If the solution is not true, then it is not a valid solution.

Q: What if I get two solutions, but neither of them is valid?

A: If you get two solutions, but neither of them is valid, then you can conclude that there are no solutions to the equation.

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 expression inside the absolute value bars
Not considering both the positive and negative cases
Not checking if the solutions are valid
Not using the correct signs when removing the absolute value bars

Q: How can I practice solving absolute value equations?

A: You can practice solving absolute value equations by working through examples and exercises in a textbook or online resource. You can also try solving absolute value equations on your own and then checking your solutions with a calculator or online tool.

Additional Resources

Khan Academy: Absolute Value Equations
Mathway: Absolute Value Equations
IXL: Absolute Value Equations

Conclusion

Solving absolute value equations can be a challenging task, but with practice and patience, you can become proficient in solving these types of equations. Remember to isolate the expression inside the absolute value bars, consider both the positive and negative cases, and check if the solutions are valid. With these tips and resources, you can master the art of solving absolute value equations and apply them to real-world problems.