Solve ∣ X − 5 ∣ + 7 = 13 |x-5|+7=13 ∣ X − 5∣ + 7 = 13 A. X = 11 X=11 X = 11 And X = − 1 X=-1 X = − 1 B. X = − 11 X=-11 X = − 11 And X = − 1 X=-1 X = − 1 C. X = 11 X=11 X = 11 And X = − 11 X=-11 X = − 11 D. X = − 11 X=-11 X = − 11 And X = 1 X=1 X = 1

Understanding Absolute Value Equations

Absolute value equations are a type of mathematical 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 this case, we are given the equation $|x-5|+7=13$, where $x$ is the variable we need to solve for.

Breaking Down the Equation

To solve the equation $|x-5|+7=13$, we need to isolate the absolute value expression. We can start by subtracting 7 from both sides of the equation, which gives us $|x-5|=6$. This equation tells us that the distance between $x$ and 5 is 6.

Solving for $x$

Now that we have the equation $|x-5|=6$, we can solve for $x$ by considering two cases:

Case 1: $x-5 \geq 0$

If $x-5 \geq 0$, then the absolute value expression $|x-5|$ is equal to $x-5$. We can substitute this into the equation $|x-5|=6$ to get $x-5=6$. Solving for $x$, we get $x=11$.

Case 2: $x-5 < 0$

If $x-5 < 0$, then the absolute value expression $|x-5|$ is equal to $-(x-5)$. We can substitute this into the equation $|x-5|=6$ to get $-(x-5)=6$. Solving for $x$, we get $x=-11$.

Checking the Solutions

We have found two possible solutions for $x$: $x=11$ and $x=-11$. To check if these solutions are correct, we can substitute them back into the original equation $|x-5|+7=13$.

For $x=11$, we have $|11-5|+7=13$, which simplifies to $6+7=13$. This is true, so $x=11$ is a valid solution.

For $x=-11$, we have $|-11-5|+7=13$, which simplifies to $16+7=13$. This is not true, so $x=-11$ is not a valid solution.

Conclusion

We have solved the equation $|x-5|+7=13$ and found that the only valid solution is $x=11$. The other options are not correct.

Final Answer

The final answer is A. $x=11$ and $x=-1$ is incorrect, the correct answer is A. $x=11$

Understanding Absolute Value Equations

Absolute value equations are a type of mathematical 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 this case, we are given the equation $|x-5|+7=13$, where $x$ is the variable we need to solve for.

Q: What is the absolute value of a number?

A: 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 two cases: one where the expression is positive and one where it is negative.

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

A: The two cases are:

• Case 1: The expression inside the absolute value is non-negative (i.e., $x-5 \geq 0$).
• Case 2: The expression inside the absolute value is negative (i.e., $x-5 < 0$).

Q: How do I solve for x in each case?

A: In Case 1, you can simply remove the absolute value and solve for x. In Case 2, you need to multiply the expression inside the absolute value by -1 and then solve for x.

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

A: If you get two different solutions for x, you need to check each solution to see if it is valid. You can do this by substituting each solution back into the original equation.

Q: What if one of my solutions is not valid?

A: If one of your solutions is not valid, you can ignore it and only consider the valid solution(s).

Q: Can I use a calculator to solve absolute value equations?

A: Yes, you can use a calculator to solve absolute value equations. However, it's always a good idea to check your work by hand to make sure you understand the solution.

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 expression
• Not considering both cases (positive and negative)
• Not checking your solutions
• Not using the correct signs when multiplying or dividing expressions

Q: How do I know if I have the correct solution?

A: You can check your solution by substituting it back into the original equation. If the equation is true, then you have the correct solution.

Q: Can I use absolute value equations in real-life situations?

A: Yes, absolute value equations can be used in a variety of real-life situations, such as:

• Measuring distances
• Calculating costs
• Modeling population growth
• Solving problems in physics and engineering

Q: What are some tips for solving absolute value equations?

A: Some tips for solving absolute value equations include:

• Read the problem carefully and understand what is being asked
• Use a systematic approach to solve the equation
• Check your work carefully
• Use a calculator to check your solutions

Q: Can I use absolute value equations to solve systems of equations?

A: Yes, you can use absolute value equations to solve systems of equations. This is known as a "system of absolute value equations."