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 \(~1~\).

Are you ready?

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) \(~4(2x-5)-2(7x+5)-x~\)

\(8x-20-14x-10-x\)
\(-7x-30\)

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