Expanding Brackets

In expanding brackets, the multiplications sign \(\scriptsize \times \) is usually not written. For instance;

\( \scriptsize 2(7a \: + \: 5a) \\ = \scriptsize 2 \: \times \: (7a \: + \: 5a) \)

2, multiplies everything inside the brackets

∴ \( \scriptsize 2(7a \: + \: 5a) \\= \scriptsize 2 (7a) \: + \: 2(5a) \\ = \scriptsize 2 \: \times \: 7a \: + \: 2 \: \times \: 5a \\ = \scriptsize 14a \: + \: 10a \\ = \scriptsize 24a \)

Example 2.4.1:

Remove the brackets and simplify the following expressions;

a. 5(x + y) 
b. -2 (p – q – r) 
c. 10x – 2(2x + 7y)
d. 2(3x – 4) – 3(5 – 2x) 
e. 5(2x – 1) + 4(x – 3) 

Solution

a. 5(x + y)

= 5 (x) + 5(y) 

= \( \scriptsize 5 \: \times \: x \: + \: 5 \: \times \: y \\ \scriptsize = 5x \: + \: 5y\)

b. -2 (p – q – r)

= -2(p) -2 (-q) -2 (-r) 

= -2(p) -2 (-q) -2 (-r) 

= -2 × p + 2 × q + 2 × r 

= -2p + 2q + 2r 

c. 10x – 2 (2x + 7y)

= 10x – 2(2x) – 2(+7y) 

= \( \scriptsize 10x \: – \: 2 \: \times \: 2x \: – \: 2 \: \times \: 7y \)

= \( \scriptsize 10x \: – \: 4x \: – \:14y \)

= \( \scriptsize 6x \: – \: 14y \)

d. 2(3x – 4) – 3(5 – 2x)

= 2(3x) + 2(-4) – 3(5) – 3(-2x)

= \( \scriptsize 2 \: \times \: 3x \: + \: 2 \: \times \: – \: 4 \: -\: 3 \: \times \: 5 \: – \: 3 \: \times \: -2x \)

= \( \scriptsize 6x \: – \: 8 \: -\: 15 \: + \: 6x \)

Collect like terms

= \( \scriptsize 6{\color{Blue} x} \: – \: 8 \: -\: 15 \: + \: 6{\color{Blue} x} \)

= \( \scriptsize 6x \: + \: 6x \: – \: 8 \: -\: 15 \)

= \( \scriptsize 12x \: – \: 23 \)

e. 5(2x – 1) + 4(x – 3) 

= 5(2x) + 5(-1) + 4(x) + 4(-3) 

= 10x – 5 + 4x – 12

Collect like terms

= 10x – 5 + 4x – 12

= 10x + 4x – 5 – 12 

= 14x – 17 

Evaluation Questions: