Enter The Values For { M, N, $}$ And { P $}$ That Complete The Difference:${ \frac{7}{x} - \frac{3}{2} = \frac{n - Mx}{px} }${$ M = \square $}$ { N = \square $}$ { P = 2 $}$

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Introduction

In this article, we will delve into the world of rational equations and explore a specific problem that requires us to find the values of mm, nn, and pp that complete the difference. Rational equations are a fundamental concept in algebra, and understanding how to solve them is crucial for success in mathematics and other fields.

The Problem

The given rational equation is:

7x−32=n−mxpx\frac{7}{x} - \frac{3}{2} = \frac{n - mx}{px}

We are also given the values of mm, nn, and pp as:

m=□m = \square

n=□n = \square

p=2p = 2

Our goal is to find the values of mm, nn, and pp that satisfy the equation.

Step 1: Simplify the Equation

To simplify the equation, we can start by finding a common denominator for the fractions on the left-hand side. The common denominator is 2x2x, so we can rewrite the equation as:

142x−32=n−mx2x\frac{14}{2x} - \frac{3}{2} = \frac{n - mx}{2x}

Step 2: Combine Like Terms

Next, we can combine the like terms on the left-hand side by finding a common denominator. The common denominator is 2x2x, so we can rewrite the equation as:

14−3x2x=n−mx2x\frac{14 - 3x}{2x} = \frac{n - mx}{2x}

Step 3: Eliminate the Denominator

Since the denominators are the same, we can eliminate the denominator by multiplying both sides of the equation by 2x2x. This gives us:

14−3x=n−mx14 - 3x = n - mx

Step 4: Rearrange the Equation

To isolate the variables, we can rearrange the equation by moving all the terms to one side. This gives us:

14−n=mx−3x14 - n = mx - 3x

Step 5: Factor Out the Common Term

We can factor out the common term xx from the right-hand side of the equation. This gives us:

14−n=x(m−3)14 - n = x(m - 3)

Step 6: Solve for mm

Since we are given the value of pp as 22, we can substitute this value into the equation. This gives us:

14−n=x(m−3)14 - n = x(m - 3)

14−n=2x(m−3)14 - n = 2x(m - 3)

14−n2x=m−3\frac{14 - n}{2x} = m - 3

m=14−n2x+3m = \frac{14 - n}{2x} + 3

Step 7: Solve for nn

We can substitute the value of mm into the equation. This gives us:

n=14−2x(m−3)n = 14 - 2x(m - 3)

n=14−2x(14−n2x+3)n = 14 - 2x\left(\frac{14 - n}{2x} + 3\right)

n=14−(14−n)−6xn = 14 - (14 - n) - 6x

n=14−14+n−6xn = 14 - 14 + n - 6x

n=n−6xn = n - 6x

6x=06x = 0

x=0x = 0

Step 8: Solve for pp

Since we are given the value of pp as 22, we can substitute this value into the equation. This gives us:

p=2p = 2

Conclusion

In this article, we have solved the rational equation and found the values of mm, nn, and pp that satisfy the equation. The values are:

m=14−n2x+3m = \frac{14 - n}{2x} + 3

n=14−2x(m−3)n = 14 - 2x(m - 3)

p=2p = 2

Note that the value of xx is 00, which means that the equation is not defined for x=0x = 0. Therefore, the values of mm and nn are not defined for x=0x = 0.

Final Answer

The final answer is:

m=14−n2x+3m = \frac{14 - n}{2x} + 3

n=14−2x(m−3)n = 14 - 2x(m - 3)

p=2p = 2

Note that the value of xx is 00, which means that the equation is not defined for x=0x = 0. Therefore, the values of mm and nn are not defined for x=0x = 0.

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Q: What is a rational equation?

A: A rational equation is an equation that contains one or more rational expressions, which are expressions that can be written in the form of a fraction.

Q: How do I simplify a rational equation?

A: To simplify a rational equation, you can start by finding a common denominator for the fractions on the left-hand side. Then, you can combine like terms and eliminate the denominator by multiplying both sides of the equation by the common denominator.

Q: What is the difference between a rational equation and a rational expression?

A: A rational expression is an expression that can be written in the form of a fraction, while a rational equation is an equation that contains one or more rational expressions.

Q: How do I solve a rational equation?

A: To solve a rational equation, you can start by simplifying the equation and then isolating the variable. You can use algebraic techniques such as adding, subtracting, multiplying, and dividing to solve for the variable.

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

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

  • Not simplifying the equation before solving
  • Not isolating the variable
  • Not checking for extraneous solutions
  • Not considering the domain of the equation

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you can substitute the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution.

Q: What is the domain of a rational equation?

A: The domain of a rational equation is the set of all possible values of the variable that make the equation true. For example, if the equation contains a fraction with a denominator of x, then the domain of the equation is all real numbers except x = 0.

Q: How do I graph a rational equation?

A: To graph a rational equation, you can start by finding the x-intercepts and y-intercepts of the equation. Then, you can use these points to graph the equation.

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

A: Rational equations have many real-world applications, including:

  • Physics: Rational equations are used to describe the motion of objects and the behavior of physical systems.
  • Engineering: Rational equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Rational equations are used to model economic systems and make predictions about economic trends.

Q: How do I use technology to solve rational equations?

A: There are many software programs and online tools that can be used to solve rational equations, including:

  • Graphing calculators
  • Computer algebra systems
  • Online equation solvers

Q: What are some common types of rational equations?

A: Some common types of rational equations include:

  • Linear rational equations
  • Quadratic rational equations
  • Polynomial rational equations
  • Rational inequalities

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you can start by simplifying the inequality and then isolating the variable. You can use algebraic techniques such as adding, subtracting, multiplying, and dividing to solve for the variable.

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

A: Some common mistakes to avoid when solving rational inequalities include:

  • Not simplifying the inequality before solving
  • Not isolating the variable
  • Not checking for extraneous solutions
  • Not considering the domain of the inequality

Q: How do I check for extraneous solutions when solving rational inequalities?

A: To check for extraneous solutions, you can substitute the solution back into the original inequality and check if it is true. If the solution is not true, then it is an extraneous solution.

Q: What is the domain of a rational inequality?

A: The domain of a rational inequality is the set of all possible values of the variable that make the inequality true. For example, if the inequality contains a fraction with a denominator of x, then the domain of the inequality is all real numbers except x = 0.

Q: How do I graph a rational inequality?

A: To graph a rational inequality, you can start by finding the x-intercepts and y-intercepts of the inequality. Then, you can use these points to graph the inequality.

Q: What are some real-world applications of rational inequalities?

A: Rational inequalities have many real-world applications, including:

  • Physics: Rational inequalities are used to describe the motion of objects and the behavior of physical systems.
  • Engineering: Rational inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Rational inequalities are used to model economic systems and make predictions about economic trends.

Q: How do I use technology to solve rational inequalities?

A: There are many software programs and online tools that can be used to solve rational inequalities, including:

  • Graphing calculators
  • Computer algebra systems
  • Online equation solvers

Q: What are some common types of rational inequalities?

A: Some common types of rational inequalities include:

  • Linear rational inequalities
  • Quadratic rational inequalities
  • Polynomial rational inequalities
  • Rational inequalities with absolute values

Q: How do I solve a rational inequality with absolute values?

A: To solve a rational inequality with absolute values, you can start by simplifying the inequality and then isolating the variable. You can use algebraic techniques such as adding, subtracting, multiplying, and dividing to solve for the variable.

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

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

  • Not simplifying the inequality before solving
  • Not isolating the variable
  • Not checking for extraneous solutions
  • Not considering the domain of the inequality

Q: How do I check for extraneous solutions when solving rational inequalities with absolute values?

A: To check for extraneous solutions, you can substitute the solution back into the original inequality and check if it is true. If the solution is not true, then it is an extraneous solution.

Q: What is the domain of a rational inequality with absolute values?

A: The domain of a rational inequality with absolute values is the set of all possible values of the variable that make the inequality true. For example, if the inequality contains a fraction with a denominator of x, then the domain of the inequality is all real numbers except x = 0.

Q: How do I graph a rational inequality with absolute values?

A: To graph a rational inequality with absolute values, you can start by finding the x-intercepts and y-intercepts of the inequality. Then, you can use these points to graph the inequality.

Q: What are some real-world applications of rational inequalities with absolute values?

A: Rational inequalities with absolute values have many real-world applications, including:

  • Physics: Rational inequalities with absolute values are used to describe the motion of objects and the behavior of physical systems.
  • Engineering: Rational inequalities with absolute values are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Rational inequalities with absolute values are used to model economic systems and make predictions about economic trends.

Q: How do I use technology to solve rational inequalities with absolute values?

A: There are many software programs and online tools that can be used to solve rational inequalities with absolute values, including:

  • Graphing calculators
  • Computer algebra systems
  • Online equation solvers

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

A: Some common types of rational inequalities with absolute values include:

  • Linear rational inequalities with absolute values
  • Quadratic rational inequalities with absolute values
  • Polynomial rational inequalities with absolute values
  • Rational inequalities with absolute values and multiple variables