Solve The Equation: $\frac{1}{2} X + 1 = |x - 2| - 1$

Introduction

Solving equations involving absolute values can be a challenging task in mathematics. The absolute value function, denoted by $|x|$, gives the distance of $x$ from zero on the number line. When solving equations with absolute values, we need to consider both the positive and negative cases of the expression inside the absolute value. In this article, we will focus on solving the equation $\frac{1}{2} x + 1 = |x - 2| - 1$.

Understanding the Equation

The given equation is $\frac{1}{2} x + 1 = |x - 2| - 1$. To solve this equation, we need to isolate the absolute value expression on one side of the equation. We can start by adding 1 to both sides of the equation, which gives us $\frac{1}{2} x + 2 = |x - 2|$.

Case 1: $x - 2 \geq 0$

When $x - 2 \geq 0$, the absolute value expression $|x - 2|$ simplifies to $x - 2$. Substituting this into the equation, we get $\frac{1}{2} x + 2 = x - 2$. To solve for $x$, we can subtract $\frac{1}{2} x$ from both sides of the equation, which gives us $2 = \frac{1}{2} x - 2$. Adding 2 to both sides of the equation, we get $4 = \frac{1}{2} x$. Multiplying both sides of the equation by 2, we get $8 = x$. Therefore, when $x - 2 \geq 0$, the solution to the equation is $x = 8$.

Case 2: $x - 2 < 0$

When $x - 2 < 0$, the absolute value expression $|x - 2|$ simplifies to $-(x - 2)$. Substituting this into the equation, we get $\frac{1}{2} x + 2 = -(x - 2)$. To solve for $x$, we can distribute the negative sign on the right-hand side of the equation, which gives us $\frac{1}{2} x + 2 = -x + 2$. Adding $x$ to both sides of the equation, we get $\frac{3}{2} x + 2 = 2$. Subtracting 2 from both sides of the equation, we get $\frac{3}{2} x = 0$. Multiplying both sides of the equation by $\frac{2}{3}$, we get $x = 0$. Therefore, when $x - 2 < 0$, the solution to the equation is $x = 0$.

Conclusion

In this article, we solved the equation $\frac{1}{2} x + 1 = |x - 2| - 1$ by considering two cases: when $x - 2 \geq 0$ and when $x - 2 < 0$. In the first case, we found that the solution to the equation is $x = 8$. In the second case, we found that the solution to the equation is $x = 0$. Therefore, the solutions to the equation are $x = 8$ and $x = 0$.

Final Thoughts

Solving equations involving absolute values requires careful consideration of the cases when the expression inside the absolute value is positive or negative. By following the steps outlined in this article, we can solve equations involving absolute values and gain a deeper understanding of the properties of absolute value functions.

Additional Resources

For more information on solving equations involving absolute values, we recommend the following resources:

• Khan Academy: Solving Equations with Absolute Values
• Mathway: Solving Equations with Absolute Values
• Wolfram Alpha: Solving Equations with Absolute Values

Frequently Asked Questions

Q: What is the absolute value function? A: The absolute value function, denoted by $|x|$, gives the distance of $x$ from zero on the number line.

Q: How do I solve an equation involving absolute values? A: To solve an equation involving absolute values, you need to consider both the positive and negative cases of the expression inside the absolute value.

Q: What are the solutions to the equation $\frac{1}{2} x + 1 = |x - 2| - 1$? A: The solutions to the equation are $x = 8$ and $x = 0$.

Q&A: Solving Equations with Absolute Values

Q: What is the absolute value function?

A: The absolute value function, denoted by $|x|$, gives the distance of $x$ from zero on the number line. It is a mathematical function that returns the non-negative value of a number, regardless of whether the number is positive or negative.

Q: How do I solve an equation involving absolute values?

A: To solve an equation involving absolute values, you need to consider both the positive and negative cases of the expression inside the absolute value. This means that you need to solve the equation for both $x - 2 \geq 0$ and $x - 2 < 0$.

Q: What are the steps to solve an equation involving absolute values?

A: The steps to solve an equation involving absolute values are as follows:

1. Isolate the absolute value expression on one side of the equation.
2. Consider the two cases: $x - 2 \geq 0$ and $x - 2 < 0$.
3. Solve the equation for each case separately.
4. Check the solutions to ensure that they satisfy the original equation.

Q: How do I know which case to use?

A: To determine which case to use, you need to consider the sign of the expression inside the absolute value. If the expression is positive, use the case $x - 2 \geq 0$. If the expression is negative, use the case $x - 2 < 0$.

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

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

• Not considering both cases: $x - 2 \geq 0$ and $x - 2 < 0$.
• Not isolating the absolute value expression on one side of the equation.
• Not checking the solutions to ensure that they satisfy the original equation.

Q: How do I check my solutions?

A: To check your solutions, you need to substitute the values back into the original equation and ensure that they satisfy the equation. This will help you to verify that your solutions are correct.

Q: What are some real-world applications of solving equations with absolute values?

A: Solving equations with absolute values has many real-world applications, including:

• Physics: To model the motion of objects with absolute values, such as the distance traveled by an object.
• Engineering: To design systems with absolute values, such as the distance between two points.
• Economics: To model the behavior of economic systems with absolute values, such as the distance between two prices.

Q: How do I practice solving equations with absolute values?

A: To practice solving equations with absolute values, you can try the following:

• Use online resources, such as Khan Academy or Mathway, to practice solving equations with absolute values.
• Work with a tutor or teacher to practice solving equations with absolute values.
• Try solving equations with absolute values on your own, using a calculator or computer to check your solutions.

Q: What are some common types of equations with absolute values?

A: Some common types of equations with absolute values include:

• Linear equations with absolute values: $\frac{1}{2} x + 1 = |x - 2| - 1$
• Quadratic equations with absolute values: $x^2 + 2x + 1 = |x - 1| - 1$
• Polynomial equations with absolute values: $x^3 + 2x^2 + x + 1 = |x - 1| - 1$

Q: How do I know which type of equation to use?

A: To determine which type of equation to use, you need to consider the degree of the polynomial and the type of absolute value expression. For example, if the polynomial is linear, use a linear equation with absolute values. If the polynomial is quadratic, use a quadratic equation with absolute values.