A Student Solves The Following Equation For All Possible Values Of $x$:$\[ \frac{8}{x+2} = \frac{2}{x-4} \\]His Solution Is As Follows:Step 1: $8(x-4) = 2(x+2)$Step 2: $4(x-4) = X+2$Step 3: \$4x - 16 =

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A Step-by-Step Guide to Solving the Equation for All Possible Values of x

Solving equations is a fundamental concept in mathematics that requires a deep understanding of algebraic manipulations. In this article, we will guide you through the process of solving a given equation for all possible values of x. The equation in question is 8x+2=2x−4\frac{8}{x+2} = \frac{2}{x-4}, and we will follow the steps taken by a student to solve it.

Step 1: Multiplying Both Sides by the Least Common Denominator

The first step in solving the equation is to eliminate the fractions by multiplying both sides by the least common denominator (LCD). In this case, the LCD is (x+2)(x−4)(x+2)(x-4). By multiplying both sides by the LCD, we get:

8(x−4)=2(x+2)8(x-4) = 2(x+2)

This step is crucial in simplifying the equation and making it easier to solve.

Step 2: Expanding and Simplifying the Equation

The next step is to expand and simplify the equation. We can do this by distributing the numbers outside the parentheses to the terms inside. This gives us:

8x−32=2x+48x - 32 = 2x + 4

Now, we can simplify the equation by combining like terms. We can subtract 2x2x from both sides to get:

6x−32=46x - 32 = 4

Step 3: Isolating the Variable

The goal of solving an equation is to isolate the variable, which in this case is x. To do this, we need to get rid of the constant term on the left-hand side. We can do this by adding 32 to both sides of the equation:

6x=366x = 36

Now, we can divide both sides by 6 to solve for x:

x=6x = 6

The student's solution is correct, but it's essential to note that the equation has a restriction on the value of x. Since the original equation has denominators of x+2 and x-4, we need to ensure that x is not equal to -2 or 4, as these values would make the denominators equal to zero.

In this case, the solution x = 6 is valid, but we need to be cautious when working with equations that have restrictions on the values of the variables.

Solving equations is a critical skill in mathematics that requires a deep understanding of algebraic manipulations. By following the steps outlined in this article, we can solve equations for all possible values of x. It's essential to be mindful of the restrictions on the values of the variables and to carefully check our work to ensure that we have found the correct solution.

Common Mistakes to Avoid

When solving equations, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not checking for restrictions: Make sure to check for restrictions on the values of the variables before solving the equation.
  • Not simplifying the equation: Simplify the equation as much as possible to make it easier to solve.
  • Not isolating the variable: Make sure to isolate the variable on one side of the equation.
  • Not checking the solution: Check the solution to ensure that it is valid and not extraneous.

By following these steps and avoiding common mistakes, we can solve equations with confidence and accuracy.

If you're struggling to solve equations or need additional practice, here are some additional resources to help you:

  • Online tutorials: Websites like Khan Academy and Mathway offer interactive tutorials and practice exercises to help you learn and practice solving equations.
  • Textbooks: Algebra textbooks like "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart provide comprehensive coverage of algebraic concepts and techniques.
  • Practice problems: Websites like IXL and Math Open Reference offer practice problems and exercises to help you reinforce your understanding of algebraic concepts.

By following these resources and practicing regularly, you can become proficient in solving equations and tackle even the most challenging problems with confidence.
A Student's Guide to Solving Equations: Q&A

Solving equations is a fundamental concept in mathematics that requires a deep understanding of algebraic manipulations. In our previous article, we guided you through the process of solving a given equation for all possible values of x. In this article, we will address some common questions and concerns that students may have when solving equations.

Q: What is the first step in solving an equation?

A: The first step in solving an equation is to eliminate the fractions by multiplying both sides by the least common denominator (LCD). This is done to simplify the equation and make it easier to solve.

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

A: To ensure that you have the correct solution, you need to check that it satisfies the original equation. This means plugging the solution back into the original equation and verifying that it is true.

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

A: Some common mistakes to avoid when solving equations include:

  • Not checking for restrictions on the values of the variables
  • Not simplifying the equation
  • Not isolating the variable
  • Not checking the solution

Q: How do I know if an equation has restrictions on the values of the variables?

A: An equation has restrictions on the values of the variables if the denominator is equal to zero. For example, in the equation 8x+2=2x−4\frac{8}{x+2} = \frac{2}{x-4}, the denominator is x+2x+2 and x−4x-4. Therefore, the equation has restrictions on the values of x, which are x = -2 and x = 4.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, the equation 2x+3=52x + 3 = 5 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, the equation x2+4x+4=0x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a, b, and c are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a, b, and c are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. Then, you need to simplify the expression and solve for x.

Q: What are some common applications of solving equations?

A: Solving equations has many common applications in real-life situations, such as:

  • Physics: Solving equations is used to describe the motion of objects and to calculate their velocities and accelerations.
  • Engineering: Solving equations is used to design and optimize systems, such as bridges and buildings.
  • Economics: Solving equations is used to model economic systems and to make predictions about future economic trends.

Solving equations is a fundamental concept in mathematics that requires a deep understanding of algebraic manipulations. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving equations and tackle even the most challenging problems with confidence. Remember to check for restrictions on the values of the variables, simplify the equation, isolate the variable, and check the solution to ensure that you have found the correct solution.

If you're struggling to solve equations or need additional practice, here are some additional resources to help you:

  • Online tutorials: Websites like Khan Academy and Mathway offer interactive tutorials and practice exercises to help you learn and practice solving equations.
  • Textbooks: Algebra textbooks like "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart provide comprehensive coverage of algebraic concepts and techniques.
  • Practice problems: Websites like IXL and Math Open Reference offer practice problems and exercises to help you reinforce your understanding of algebraic concepts.

By following these resources and practicing regularly, you can become proficient in solving equations and tackle even the most challenging problems with confidence.