Expressions with Brackets

In AlgebraAlgebra is a branch of mathematics that substitutes letters for numbers. Algebra is about finding the unknown or putting real-life variables into equations and then solving them. More, letters stand for numbers. The number can be whole, positive or negative. To solve algebraic expressions containing brackets, we solve the brackets first e.g. 

a. \( \scriptsize 5x \: + \: (4x \: – \: 3x) \)

The bracket contains the expression 4x – 3x which is equal to x

4x – 3x = x

We can then solve the whole equation since we have solved what was in the brackets

∴ \( \scriptsize 5x \: + \: (4x \: – \: 3x) \\ \scriptsize = 5x \: + \: x \\ = \scriptsize 6x \)

b. \( \scriptsize 9x \: – \: (5x \: + \: 2x)\)

The bracket contains the expression 5x + 2x which is equal to 7x

5x + 2x = 7x

We can then solve the whole equation since we have solved what was in the brackets

∴ \( \scriptsize 9x \: – \: (5x \: + \: 2x) \\ \scriptsize 9x \: – \: 7x \\ = \scriptsize 2x \)

Rules for Removing Brackets in Algebraic:

(i) “+ sign“ before a bracket multiplies everything within the bracket and hence does not change the sign of terms in the bracket e.g. 

+(5 + 6) = +5 + 6

(ii) “– sign” before a bracket multiples every term within the bracket, and hence changes the sign of all the terms in the bracket i.e. “+ changes to –” and “– changes to +“,

-(3)      +3 changes to -3

-(-5)      -5 changes to +5

If there is no sign in front of a number it typically means that the number is positive. For example, +3 is positive, −3 is negative, and 3 is positive.

e.g.

9x – (5x + 2x) = 9x – (5x) – (+2x) 

= 9x – 5x – 2x

= 9x – 7x

= 2x. 

Note that +5x changed to -5x
+2x changed to -2x
Remember the sign convention 

Example 2.1.1:

Simplify the following by removing the brackets; 

a. 5x + (7x – 3x)
b. m + (n – p)
c. 3a + (2b + a)
d. 3(x + y)
e. 3q (-p + q)
f. 4p – (-8p – 3q) 
g. m – (n – p) 
h. 3a – (2b + a) 
i. -3 (-x + y) 
j. -3q (-p + q) 

Solution 

a. 5x + (7x – 3x)

= 5x + (7x) + (– 3x)

= 5x + 7x – 3x

= 12x – 3x = 9x

b. m + (n – p)

= m + (n) + (-p) 

= m + n – p

c.  3a + (2b + a) = 3a + (2b) + (+a) 

 = 3a + 2b + a

collect like terms

 = 3a + 2b + a

∴ 3a + a + 2b

= 4a + 2b

d. 3(x + y) = 3(x)  + 3(y) 

= 3x + 3y 

e. 3q (-p + q)

= \( \scriptsize 3q(-p) \: + \: 3q(q) \)

= \( \scriptsize 3q \: \times \: -p \: + \: 3q \: \times \: q \)

= \( \scriptsize -3pq + \: 3q^2 \)

or \( \scriptsize 3q^2\: – \: 3pq \)

f. 4p – (-8p – 3q)

= 4p – (-8p) – (-3q) 

= 4p + 8p + 3q 

Like terms have already been collected

4p + 8p + 3q 

= 4p + 8p + 3q

= 12p + 3q 

g. m – (n – p)

= m – (n) – (-p)

=   m – n + p 

h. 3a – (2b + a)

= 3a – (2b) – (+ a)

= 3a – 2b – a 

Collect like terms

= 3a – 2b – a 

= 3a – a – 2b

= 2a – 2b 

  i. -3 (-x + y)

-3 (-x)  – 3 (+ y)

= +3x – 3y

= 3x – 3y

j. -3q(-p + q) 

=  \( \scriptsize -3q(-p) \: -3q (+q) \)

= \( \scriptsize -3q \: \times \: -p \: -3q \: \times \: +q \)

= \( \scriptsize +3qp \: – \: 3q^2 \)

Evaluation Questions: