## Simplifying Rational Expressions

#### Steps to solve a rational equation algebraically

1. Factorize the polynomials in the denominator and numerator.

2. Cancel the common factors between the numerator and denominator.

3. There are some important formulas for factorizing polynomials. Just like for quadratic equation we know that (x+a)2 = x2+a2+2ax. Some of these formulas are given below:

We can check these formulas by multiplying the factors using the distributive property.

1. (x+a)3 = x3 + a3 + 3x2a + 3ax2

2. (x-a)3 = x3 - a3 - 3x2a + 3ax2

3. (x3+a3) = (x+a)(x2 + a2 - ax)

Let's try to prove it:

Right hand side = (x+a)(x2 + a2 - ax)

= x3 + xa2 – ax2 + ax2 + a3 – a2x

= x3 + a3 = Left hand side = proof.

4. (x3 - a3) = (x - a)(x2 + a2 + ax)

5. Example 1: Simplify

Solution = #### Solving Rational Equations and Inequalities Algebraically

1. We follow the same steps as mentioned above to simplify the rational expression.

2. The simplified expression is then solved to find the value of x in the equation given by r1(x) = r2(x) or r(x)
= a where r(x), r1(x) and r2(x) are any rational expressions and 'a' is a constant. We use the cross multiplication
method to solve the equation.

3. Example : 1

1. The expressions are already simplified.

2. Cross multiply, i.e. multiply the numerator of the left side by the denominator of the right side = the
numerator of the right side multiplied by the denominator of the left side.

• 3x=4x+4

• Move the variables to the left side and the constants to the right side by using inverse operations.

• Subtract 4x from both the sides.

• 3x-4x=4

• -x = 4 or x = -4.

Example : 2

The same question above can be changed to an inequality as shown: We use the same steps as for example 1:

3x ≥ 4x+4

• -x ≥ 4

• Multiplying both the sides with -1, x ≤ -4.

#### Try these problems

1. Solve the rational equation 9/28 + 3/(z + 2) = 3 / 4.

1. -5

2. 4

3. -4

4. 5 Thus, the answer = 