Determine The Solutions Of The Equation:${ \left|\frac{1}{4} X + 7\right| - 3 = 24 }$A. { X = -136$}$ And { X = 136$}$B. { X = -136$}$ And { X = 80$}$C. { X = -112$}$ And [$x =

by ADMIN 178 views

Introduction

In mathematics, absolute value equations are a type of equation that involves the absolute value of a variable or expression. These equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore how to solve absolute value equations, using the equation ∣14x+7∣−3=24\left|\frac{1}{4} x + 7\right| - 3 = 24 as a case study.

Understanding Absolute Value Equations

Absolute value equations involve 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. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5.

When solving absolute value equations, we need to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative. This is because the absolute value of a number is always non-negative.

Step 1: Isolate the Absolute Value Expression

To solve the equation ∣14x+7∣−3=24\left|\frac{1}{4} x + 7\right| - 3 = 24, we need to isolate the absolute value expression. We can do this by adding 3 to both sides of the equation:

∣14x+7∣=24+3\left|\frac{1}{4} x + 7\right| = 24 + 3

This simplifies to:

∣14x+7∣=27\left|\frac{1}{4} x + 7\right| = 27

Step 2: Set Up Two Equations

Now that we have isolated the absolute value expression, we can set up two equations: one where the expression inside the absolute value is positive, and one where it is negative.

Let's start with the case where the expression inside the absolute value is positive:

14x+7=27\frac{1}{4} x + 7 = 27

We can solve this equation by subtracting 7 from both sides:

14x=20\frac{1}{4} x = 20

Multiplying both sides by 4 gives us:

x=80x = 80

Now, let's consider the case where the expression inside the absolute value is negative:

14x+7=−27\frac{1}{4} x + 7 = -27

We can solve this equation by subtracting 7 from both sides:

14x=−34\frac{1}{4} x = -34

Multiplying both sides by 4 gives us:

x=−136x = -136

Step 3: Check the Solutions

Now that we have found two possible solutions, we need to check them to make sure they are valid. We can do this by plugging each solution back into the original equation.

For x=80x = 80, we have:

∣14(80)+7∣−3=∣20+7∣−3=27−3=24\left|\frac{1}{4} (80) + 7\right| - 3 = \left|20 + 7\right| - 3 = 27 - 3 = 24

This confirms that x=80x = 80 is a valid solution.

For x=−136x = -136, we have:

∣14(−136)+7∣−3=∣−34+7∣−3=27−3=24\left|\frac{1}{4} (-136) + 7\right| - 3 = \left|-34 + 7\right| - 3 = 27 - 3 = 24

This confirms that x=−136x = -136 is also a valid solution.

Conclusion

In this article, we have explored how to solve absolute value equations using the equation ∣14x+7∣−3=24\left|\frac{1}{4} x + 7\right| - 3 = 24 as a case study. We have seen how to isolate the absolute value expression, set up two equations, and check the solutions. With practice and patience, you can master the art of solving absolute value equations and tackle even the most challenging problems with ease.

Final Answer

Introduction

In our previous article, we explored how to solve absolute value equations using the equation ∣14x+7∣−3=24\left|\frac{1}{4} x + 7\right| - 3 = 24 as a case study. In this article, we will answer some of the most frequently asked questions about absolute value equations.

Q: What is an absolute value equation?

A: An absolute value equation is a type of 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.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative. You can then set up two equations and solve for the variable.

Q: What is the difference between an absolute value equation and a linear equation?

A: An absolute value equation involves the absolute value of a variable or expression, while a linear equation involves a linear expression. For example, the equation ∣x∣=5|x| = 5 is an absolute value equation, while the equation x+2=7x + 2 = 7 is a linear equation.

Q: Can I use algebraic manipulations to solve absolute value equations?

A: Yes, you can use algebraic manipulations to solve absolute value equations. For example, you can add or subtract the same value from both sides of the equation, or multiply or divide both sides by the same non-zero value.

Q: How do I check my solutions to an absolute value equation?

A: To check your solutions to an absolute value equation, you need to plug each solution back into the original equation and verify that it is true. This will help you ensure that your solutions are valid.

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:

  • Failing to consider both cases (positive and negative) when solving the equation
  • Not checking the solutions to ensure they are valid
  • Not using algebraic manipulations correctly to solve the equation
  • Not being careful with signs and directions when working with absolute values

Q: Can I use technology to solve absolute value equations?

A: Yes, you can use technology to solve absolute value equations. For example, you can use a graphing calculator or a computer algebra system to solve the equation and find the solutions.

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

A: Absolute value equations have many real-world applications, including:

  • Physics: to model the motion of objects with constant acceleration
  • Engineering: to design and optimize systems with constraints
  • Economics: to model the behavior of economic systems with uncertainty
  • Computer Science: to solve problems involving distances and directions

Conclusion

In this article, we have answered some of the most frequently asked questions about absolute value equations. We hope that this article has provided you with a better understanding of how to solve absolute value equations and has helped you to avoid common mistakes. With practice and patience, you can master the art of solving absolute value equations and tackle even the most challenging problems with ease.

Final Answer

The final answer is: A\boxed{A}