ADDING AND SUBTRACTING RATIONAL EXPRESSIONS

Example 1 :

Simplify :

[(x + 2)/(x + 3)] + [(x - 1)/(x + 3)]

Solution :

=  [(x + 2)/(x + 3)] + [(x - 1)/(x + 3)]

Since the denominators are same, take the denominator once and combine the numerators. 

=  [(x + 2) + (x - 1)]/(x + 3)]

=  (x + 2 + x - 1)/(x + 3)

=  (2x + 1)/(x + 3)

Example 2 :

Simplify : 

[(x + 1)/(x - 1)²] + [1/(x + 1)]

Solution :

=  [(x + 1)/(x - 1)²] + [1/(x + 1)]

Since the denominators are not the same, find the LCM.

The LCM of (x - 1)² and (x + 1) is (x - 1)²(x + 1).

Rewrite the expressions using the LCM.

First expression : 

=  (x + 1)/(x - 1)²

=  [(x + 1)(x + 1)]/[(x - 1)²(x + 1)]

=  (x + 1)²/(x - 1)²(x + 1)

Second expression :

=  1/(x + 1)

=  1(x - 1)²/[(x + 1)(x - 1)²]

=  (x - 1)²/(x - 1)²(x + 1)

Add :

=  [(x + 1)²/(x - 1)²(x + 1)] + [(x - 1)²/(x - 1)²(x + 1)]

=  [(x + 1)² + (x - 1)²] / [(x - 1)²(x + 1)]

=  [x² + 2x + 1 + x² -2x + 1] / [(x - 1)²(x + 1)]

=  (2x² + 2) / [(x - 1)²(x + 1)]

Example 3 :

Simplify :

[(4x + 3)/(x² + 3x + 2)] + [(x + 2)/(x² + 3x + 2)]

Solution :

=  [(4x + 3)/(x² + 3x + 2)] + [(x + 2)/(x² + 3x + 2)]

Since the denominators are same, take the denominator once and combine the numerators. 

[(4x + 3) + (x + 2)]/(x² + 3x + 2)

=  (4x + 3 + x + 2)/(x² + 3x + 2)

=  (5x + 5)/(x² + 3x + 2)

Factor the numerator and denominator. 

=  5(x + 1)/[(x + 1)(x + 2)]

Cancel common factors. 

=  5/(x + 2)

Example 4 :

Simplify :

1/(x² + 5x + 6) - 1/(x² + 6x + 8)

Solution :

=  1/(x² + 5x + 6) - 1/(x² + 6x + 8)

The denominators are not same. To find LCM of the denominators, factor both the denominators. 

=  1/[(x + 2)(x + 3)] + 1/[(x + 2)(x + 4)]

The LCM of denominators is (x + 2)(x + 3)(x + 4). 

Rewrite the expressions using the LCM.

First expression : 

=  1/[(x + 2)(x + 3)]

=  (x + 4)]/[(x + 2)(x + 3)(x + 4)]

Second expression :

=  1/(x + 2)(x + 4)

=  (x + 3)/[(x + 2)(x + 3)(x + 4)]

Now, the denominators are same. Take the denominator once and subtract the numerators. 

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

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

=  1/[(x + 2)(x + 3)(x + 4)]

Example 5 :

Simplify :

[x³/(x - 2)] + [8/(2 - x)]

Solution :

=  [x³/(x - 2)] + [8/(2 - x)]

Make the denominators same. 

=  [x³/(x - 2)] + [-8/(x - 2)]

Since the denominators are same, take the denominator once and combine the numerators. 

=  [x³ + (-8)]/(x - 2)

=  (x³ - 8)/(x - 2)

=  (x³ - 2³)/(x - 2)

The numerator is in the form of (a³ - b³), factor (x³ - 2³) using the identity shown below. 

a³ - b³ = (a - b)(a² + ab + b²)

Then, 

=  [(x - 2)(x² + 2x + 2²)]/(x - 2)

=  [(x - 2)(x² + 2x + 4)]/(x - 2)

Cancel common factors. 

=  (x² + 2x + 4)

Example 6 :

Simplify :

a³/(a - b) - b³/(a - b)

Solution :

=  a³/(a - b) - b³/(a - b)

Since the denominators are same, take the denominator once and combine the numerators. 

=  (a³ - b³)/(a - b)

Factor the numerator.

=  (a - b)(a² + ab + b²)/(a - b)

Cancel common factors.

=  a² + ab + b²

Example 7 :

Simplify :

x²/(x⁴ - y⁴) - y²/(x⁴ - y⁴)

Solution :

=  x²/(x⁴ - y⁴) - y²/(x⁴ - y⁴)

The denominators are not same. To get LCM of denominators, factor both the denominators. 

=  (x² - y²)/(x⁴ - y⁴)

=  (x² - y²)/[(x²)² - (y²)²]

Let a = x² and b = y².

=  (a - b)/(a² - b²)

Factor denominator. 

=  (a - b)/[(a + b)(a - b)]

Cancel common factors.

=  1/(a + b)

Replace a by x² and by y².

=  1/(x² + y²)

Example 8 :

If P = x/(x + y) and Q = y/(x + y) find

[1/(P - Q)] - [2Q/(P² - Q²)]

Solution :

P - Q :

=  x/(x + y) - y(x + y)

=  (x - y)/(x + y)

1/(P - Q) : 

=  (x + y)/(x - y)

P² - Q² :

=  [x/(x + y)]² - [y(x + y)]²

=  [x²/(x + y)²] - [y²(x + y)²]

=  (x² - y²)/(x + y)²

=  [(x + y)(x - y)]/(x + y)²

=  (x - y)/(x + y)

2Q :

= 2y/(x + y)

2Q/(P² - Q²) :

=  [2y/(x + y)] ÷ [(x - y)/(x + y)]

=  [2y/(x + y)] ⋅ [(x + y)/(x - y)]

=  [2y(x + y)] / [(x + y)(x - y)]

=  2y(x - y)

[1/(P - Q)] - [2Q/(P² - Q²)] :

=  [(x + y)/(x - y)] - [2y(x - y)]

=  [(x + y) - 2y] / (x - y)

=  (x + y - 2y) / (x - y)

=  (x - y)/(x - y)

=  1

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