# Lesson 9: Expanding single brackets & simplifying

Introduction

You're not really being hit by anything brand new this lesson. Just carefully apply what you learned in lessons 7 and 8 and you'll be fine. Beware, though, there is one bit that catches loads of people out!

SECTION A

Let's start with the basics. Expand the brackets first (Lesson 8) then collect like terms (Lesson 7).

Example 1

Expand and simplify $$~2(3x+5)+7~$$

Expand the bracket ⇒ $$~6x+10+7~$$

Collect like terms ⇒ $$~6x+17~$$

Example 2

Expand and simplify $$~7(x+2)-2x~$$

Expand the bracket ⇒ $$~7x+14-2x~$$

Collect like terms ⇒ $$~5x+14~$$

Example 3

Expand and simplify $$~4(5x-4)+5(2x+1)~$$

Expand the brackets ⇒ $$~20x-16+10x+5~$$

Collect like terms ⇒ $$~30x-11~$$

Example 4

Expand and simplify $$~2(6x-1)+(4x+3)~$$

Write in the invisible $$~1~$$ (see below) ⇒ $$~2(6x-1)+1(4x+3)~$$

Expand the brackets ⇒ $$~12x-2+4x+3~$$

Collect like terms ⇒ $$~16x+1~$$

In this example, writing in the invisible $$~1~$$ is not really necessary BUT keep it in mind for Section B!

The first set of practice questions will get you into the swing of things.

# Practise to master

SECTION A

Expand and simplify the following expressions.

01) $$~3(x+4)+6~$$

$$3x+12+6$$
$$3x+18$$

02) $$~8(2x+6)-2~$$

$$16x+48-2$$
$$16x+46$$

03) $$~2(7x-9)+5~$$

$$14x-18+5$$
$$14x-13$$

04) $$~5(4x-3)-1~$$

$$20x-15-1$$
$$20x-16$$

05) $$~6(3x+2)-7x~$$

$$18x+12-7x$$
$$11x+12$$

06) $$~4(9x-1)+x~$$

$$36x-4+x$$
$$37x-4$$

07) $$~7(3x+4)+9(x+3)~$$

$$21x+28+9x+27$$
$$30x+55$$

08) $$~5(2x+8)+3(7x+7)~$$

$$10x+40+21x+21$$
$$31x+61$$

09) $$~2(6x-5)+(9x+8)~$$

$$12x-10+9x+8$$
$$21x-2$$

10) $$~4(5x+3)+7(3x-4)~$$

$$20x+12+21x-28$$
$$41x-16$$

11) $$~8(4x-2)+(5x-5)~$$

$$32x-16+5x-5$$
$$37x-21$$

12) $$~6(7x-9)+5(4x-6)~$$

$$42x-54+20x-30$$
$$62x-84$$

SECTION B

Remember I said there's one bit that catches loads of people out? Well, it's a comin'!

Example 1

Expand and simplify $$~6(2x-1)-2(5x-4)~$$

Expand the brackets ⇒ $$~12x-6-10x+8~$$

Collect like terms $$~2x+2~$$

Careful! Our first step is to multiply everything in the second bracket by MINUS $$~2~$$. In Section A, this 'middle' sign was always positive so we didn't really think about it. If it's negative, we think about it!

Example 2

Expand and simplify $$~2(8x+3)-(3x+7)~$$

Write in the invisible $$~1~$$ (see below) ⇒ $$~2(8x+3)-1(3x+7)~$$

Expand the brackets ⇒ $$~16x+6-3x-7~$$

Collect like terms $$~13x-1~$$

In this example, writing in the invisible $$~1~$$ is definitely recommended AND both terms in the second bracket must be multiplied by MINUS $$~1~$$.

# Practise to master

SECTION B

Expand and simplify the following expressions.

01) $$~7(5x+1)-6(2x+9)~$$

Multiply both terms in the second bracket by MINUS $$~6~$$
$$35x+7-12x-54$$
$$23x-47$$

02) $$~4(9x+7)-3(4x+4)~$$

Multiply both terms in the second bracket by MINUS $$~3~$$
$$36x+28-12x-12$$
$$24x+16$$

03) $$~5(8x-4)-(3x-3)~$$

Write in the invisible $$~1~$$ ⇒ $$~5(8x-4)-1(3x-3)~$$
Multiply both terms in the second bracket by MINUS $$~1~$$
$$40x-20-3x+3$$
$$37x-17$$

04) $$~3(5x+8)-9(2x-5)~$$

$$15x+24-18x+45$$
$$-3x+69$$ or $$69-3x$$

05) $$~2(6x-5)-(5x+2)~$$

Write in the invisible $$~1~$$ ⇒ $$~2(6x-5)-1(5x+2)~$$
$$12x-10-5x-2$$
$$7x-12$$

06) $$~6(2x+2)-4(6x-6)~$$

$$12x+12-24x+24$$
$$-12x+36$$ or $$36-12x$$

07) $$~5(4x+9)+3(6x+6)+4~$$

$$20x+45+18x+18+4$$
$$38x+67$$

08) $$~7(6x-1)-7(4x+4)+3x~$$

$$42x-7-28x-28+3x$$
$$17x-35$$

09) $$~3(8x+3)+9(2x-2)-12~$$

$$24x+9+18x-18-12$$
$$42x-21$$

10) $$~2(5x-6)-(3x-1)+7x~$$

Write in the invisible $$~1~$$ ⇒ $$2(5x-6)-1(3x-1)+7x$$
$$10x-12-3x+1+7x$$
$$14x-11$$

11) $$~6(3x+2)+4(5x+8)+7~$$

$$18x+12+20x+32+7$$
$$38x+51$$

12) $$~4(2x-5)-2(7x+5)-x~$$

$$8x-20-14x-10-x$$
$$-7x-30$$