Solving Rational Equations | General Mathematics

Summary

This educational video tutorial guides viewers through solving rational equations, which are equations containing rational expressions, or quotients of two polynomials. The presenter methodically demonstrates the process using three examples, starting with finding the least common denominator (LCD), multiplying through by the LCD, simplifying, and solving for the variable. Each solution is checked for extraneous roots. The video is interactive, encouraging viewers to ask questions in the comments section, and the presenter ensures clarity by checking solutions in the original equations.

Takeaways

• 📘 The video is a tutorial on solving rational equations, which are equations involving rational expressions.
• 🔢 A rational expression is defined as a quotient of two polynomials.
• ❌ The instructor emphasizes that the denominator should not be equal to zero.
• 📝 The first example involves solving the equation 2/x - 3/2x = 1/5 by finding the least common denominator (LCD).
• 🔄 The process of solving involves multiplying each term by the LCD and simplifying the equation.
• 🔍 After simplifying, the example leads to solving for x, which results in x = 5/2.
• 🔄 The solution x = 5/2 is checked by substituting it back into the original equation to confirm its validity.
• 📐 The second example demonstrates solving a rational equation with binomials in the denominator.
• 🔢 The LCD for the second example is the product of the binomials (x + 1)(x - 3), and the solution is found to be x = 11.
• 🔄 The solution x = 11 is verified by substitution, confirming it is a valid solution to the equation.
• 📘 The third example is a more complex rational equation that simplifies into a quadratic equation, leading to two potential solutions, x = -7 and x = 2.
• 🔄 The solutions are checked, and it's found that x = -7 is valid while x = 2 is an extraneous solution due to making the denominator zero.

Q & A

What is a rational equation according to the video?

A rational equation is an equation involving rational expressions, which are expressions that can be written as a quotient of two polynomials.

Why is it important to not set the LCD to zero when solving rational equations?

Setting the LCD (Least Common Denominator) to zero would eliminate the denominators, which is not allowed in mathematics as it can lead to undefined expressions and loss of potential solutions.

What is the first step in solving the example equation 2/(x-3) + 3/2 = 1/5?

The first step is to find the LCD, which in this case is 10x, by multiplying the denominators x, 2, and 5.

How is the equation 2/(x-3) + 3/2 = 1/5 transformed after multiplying by the LCD?

After multiplying by the LCD (10x), the equation becomes 20x/x + 30x/2x = 10x/5.

What is the simplified form of the equation after the first example's transformation?

The simplified form is 20x = 15, which simplifies further to x = 5/2 after combining like terms and solving for x.

How do you check if the solution x = 5/2 is valid for the first example?

You substitute x = 5/2 back into the original equation and simplify to ensure both sides of the equation are equal.

What is the LCD for the second example equation 3/x + 1 = 2/(x-3)?

The LCD for the second example is the product of the two binomials (x + 1) and (x - 3).

How is the equation 3/x + 1 = 2/(x-3) simplified after multiplying by the LCD?

After multiplying by the LCD, the equation simplifies to 3(x - 3) = 2(x + 1), which further simplifies to 3x - 9 = 2x + 2.

What is the solution for x in the second example after simplification?

After simplifying and solving for x, the solution is x = 11.

How do you verify the solution x = 11 in the second example?

You substitute x = 11 back into the original equation and check if both sides are equal, confirming it is a valid solution.

What is the LCD for the third example equation x^2 - 4x / (x - 2) = 14 - 9x / (x - 2)?

The LCD for the third example is x - 2, as it is the common denominator in the equation.

What type of equation results after multiplying the third example by the LCD?

After multiplying by the LCD, the resulting equation is a quadratic equation: x^2 - 4x = 14 - 9x.

What are the solutions for x in the third example after solving the quadratic equation?

The solutions for x are x = -7 and x = 2, found by factoring the quadratic equation.

How do you determine if x = 2 is an extraneous solution in the third example?

By substituting x = 2 into the original equation, you find that the denominator becomes zero, making it an extraneous solution.