Solving Rational Equations

Algebra, a branch of mathematics for, solving rational equations, deals with the symbols and arithmetic operations related to those symbols. This branch of mathematics covers a vast syllabus in itself. Suppose one needs to divide algebra into sections.

In that case, there are various sections like Polynomials, Linear equations in one variable, Linear equations in two variables, Quadratic equations, Arithmetic progressions, Rational expressions, Exponents, and many more.

Rational expressions are counted under Elementary Algebra and are one of the important matters. It is one of the most important topics in Mathematics. Do you know about rational equations? No! Then, this article is exclusively only for you.

In this article, you will learn about solving rational equations and other terms related to them.

What is Rational Expression?

An expression that is written in the form of numerator and denominator, i.e., in the form of p(x)/q(x), where denominator q(x)≠0, is known as a rational expression.

Do you know?

Every polynomial is also a rational expression. If p(x) is any polynomial, then we can write p(x) = P(x)/1, which is a form of rational expression.

Here are some examples of radical expressions:

Suppose p(x) and q(x) are polynomials with integral coefficients such that the HCF of p(x) and q(x) is 1. We can say that p(x)q(x)is a rational expression in its lowest terms.

What is Rational Equation?

A rational equation can be defined as an equation that involves at least one rational expression, or you can say a fraction. For example,

2x+1/4 = x/3 is a rational equation.

These fractions can be on one side or both sides of the equation. It can also be a ratio of two polynomials.

A rational expression can be reduced to the lowest terms if its numerator and denominator are reduced to the lowest.

For example,

p(x) = (x+3)(x-1)/x(x+3) = (x-1)/x

The numerator and denominator of p(x) can not be reduced anymore. Therefore, it is the lowest term of p(x).

How to solve Rational Expressions?

Solving rational expressions can be performed by applying arithmetic operations like addition, subtraction, multiplication, division, etc.

Below are the methods for solving rational expressions using these operations:

1. Addition operations:

Addition of rational expressions is define as,

• p(x)/q(x) + r(x)/q(x) = p(x) + r(x) / q(x)
• p(x)/q(x) + r(x)/s(x) = p(x)/s(x) + r(x)q(x) / q(x)s(x)

2. Subtraction operations:

Subtraction of rational expressions is define as,

• p(x)/q(x) – r(x)/s(x) = p(x)/s(x) – r(x)q(x) / q(x)s(x)

3. Multiplication operations:

Multiplication of rational expressions is define as,

• p(x)/q(x) x r(x)/s(x) = p(x).r(x) / q(x).s(x)

4. Division operations:

Division of rational expressions is define as,

• p(x)/q(x)÷r(x)/s(x) = p(x)/q(x) x s(x)/r(x) = p(x).s(x)/q(x).r(x)

How to solve Rational Equations?

Solving rational equations implies getting the value of variables present in the rational equation. Now, the question arises of how to solve rational equations? Solving rational equations means clearing fractions by multiplying both sides of the equation by the least common multiple of the denominator. Remember, the denominator should not be zero.

There are many methods for solving rational equations. Solution by LCD method, Factorization method, solving by LCM method, cross-multiplying method, etc. Solving rational equations calculators are also used nowadays to solving rational equations.

Solving rational equation calculators are online tools used to find the solution of the variable in the given rational equation. Suppose you want to find the solution to the rational equation using an online calculator. You need to enter the numerator and denominator in the input box and then click on the simplified option.

Steps for solving Rational equations by LCD method are as follows:

Step 1: Factorize all the denominators to find the least common denominator.

Step 2: Multiply both sides of the equation, i.e., every term of the equation with the least common denominator.

Step 3: Solve the resulting equation.

Step 4: If you get extraneous solutions, check the solutions and reject all those values resulting in the zero denominators in the original equation.

For example,

Solve 1/x + 2/x² = x+9/2x².

Solution:

Step 1: Find the LCD.

On both sides of the equation, the denominators are x, x², and 2x².

So, the LCD we get is 2x².

Step 2: Multiplying LCD with both sides of equation.

2x² (1/x + 2/x²) = 2x² (x+9/2x²)

Step 3: Solving the resultant equation.

[2x². 1/x] + [2x². 2/x²] = 2x² (x+9/2x²)

2x + 4 = x + 9

2x – x = 9 – 4

x = 5

Therefore, x = 5 is the solution for  1/x + 2x² = x+9/2x².

Some Examples of How to Solve Rational Equations 

Here are some examples that will help you understand the method for solving rational equations.

Example 1: Solve the equation

3(x+2)/(x-4) – (x+4)/(x-2) = (x-2)/(x-4).

Solution:

Step 1: Find LCD.

Denominators are (x-4) and (x-2). So, the LCD in this case is (x-4)(x-2).

Step 2: Multiplying LCD with each term of the equation.

[(x – 4).(x – 2).3(x+2)/(x-4)] – [(x – 4).(x – 2).(x+4)/(x-2)] = [(x – 4).(x – 2).(x-2)/(x-4)]

[(x – 2).3(x + 2)] – [(x – 4).(x+4)] = [(x – 2).(x – 2)]

Step 3: Solve the resulting equation.

[3(x² – 4)] – [x² – 16] = [x² + 4 – 4x]

3x² – 12 – x² + 16 = x² – 4x + 4

2x² + 4 = x² – 4x + 4

2x² – x² + 4x = 4 – 4

x² + 4x = 0

x(x + 4) = 0

x = 0, -4

Therefore, x = {0, -4} are the solutions for 3(x+2)/(x-4) – (x+4)/(x-2) = (x-2) / (x-4).

Example 2: Solve the following equation

2/3 = x/3.

Solution:

Step 1: Find LCD.

In this case, the denominator of both sides is the same. Hence, LCD is 3.

Step 2: Multiply both sides with LCD.

3 (2/3) = 3(x/3)

Step 3: Solve the resulting equation.

2 = x

Or, x = 2

Therefore, x = 2 for the given rational equation.

How to Solve Equations with Rational Exponents?

Radical expressions, they can be written without using the radical symbol. Rational or fractional exponents are used in those cases to denote radical expressions.

For example,

n√x = x^1/n

Here n√ is called a radical symbol, while 1/n is a rational exponent. A rational exponent indicates power in the numerator and a root in the denominator.

The steps to solve equations with rational exponents are as follows:

Step 1: Isolate the expression with a rational exponent.

Step 2: Raise both sides of the equation to the reciprocal power.

Step 3: Solve for the value of the variable.

For example,

Solve x^2/3 = 64.

Solution:

In this case, the rational exponent is already isolated. So, we move to the next step.

Step 1: Raising the power of both sides.

(x^2/3)^3/2 = (64)^3/2

Step 2: Solving for the value of a variable.

x = (64)^3/2

x = [(8²)^3/2]

x = 8^{2x 3/2}

x = 8³

x = 512

Therefore, x = 512 for x^2/3 = 64.

How to Solve Equations with Rational Coefficients?

In simple words, the coefficient is defined as the number or term multiplied by the variable. In the case of the rational coefficient, the coefficient multiplied by the variable must be in the form of a fraction.

For example,

2/3 x = 8

Here, 2/3 is the rational coefficient of x.

To solve equations with rational coefficients, you need to follow these steps:

Step 1: Isolate the expression with a rational coefficient.

Step 2: Find the LCD of the equation.

Step 3: Solve the variable.

Example: Solve 7/10 n + 3/2 = 3/5 n + 3.

Solution:

Step 1: Isolate the expressions with a rational coefficient.

7/10 n – 3/5 n = 3 – 3/2

Step 2: Find the LCD.

Here, denominators are 10, 5, 1, and 2. So, the LCD is 10.

Step 3: Solve the value for the variable.

10(7/10 n ) – 10(3/5 n) = 10(3) – 10(3/2)

7n – 6n = 30 – 15

n = 15

Therefore, n = 15 for 7/10 n + 3/2 = 3/5 n + 3.

Solving Rational Equations Worksheet

Solving rational equations worksheets will help you understand the topic more precisely. To practice more on this topic, you can try the following given problems.

1. If 5x/x+2 = 7, then solve for x.
2. Combine 2/x+4 + 3/x-1 into a single rational expression.
3. Solve for x, x²-5x+6 / x²+3x+2 = 0

4. Solve the following equation: x-3/7 = 4x+12/7

5. If 3/x+2 – 1/x = 1/5x, then solve for x.
6. If 3/x+4 + 7/x-4 = 8/x²-16, then solve for x.
7. Simplify 9x = 18.

Frequently Asked Questions

1. How do you solve rational equations?

There are a few different ways to solve rational equations.

The first is to divide both sides by the same denominator.

If the equation is ax + by = c, then you can divide all three terms by a, which will result in x + y = c/a.

If you are ever unsure whether or not to divide both sides by a common denominator, just try it and see what happens! You’ll never know if you don’t try.

2. What is the easiest way to solve a rational equation?

The easiest way to solve a rational equation is to factor the denominator into primes and then multiply the top and bottom by those primes, using FOIL.

3. What are 5 examples of rational equations?

Here are 5 examples of rational equations:

x/y=3

2y/4x=6

3x/(y-2)=-1

(x+7)/(y-4)=5

2x/(y*z)=(x+z)

4. What are rational equations used for?

Rational equations are used for the following:

-Solving for an unknown variable

-Determining the relationship between two variables

-Finding the value of one variable when given the other variable

5. How do you identify rational equations?

You can identify rational equations by looking at them to see if they contain fractions. If they do, then you know that it is a rational equation.