10-Week Mathematics Catch-Up Programme (CAPS)

Week 2 – Day 3

This lesson marks another important step towards Grade 10 and Grade 11 mathematics. Two topics that many learners find difficult—inequalities and changing the subject of a formula—are introduced in Grade 9 and then used extensively in later grades.

The goal today is not mastery, but to become comfortable with the ideas.

Inequalities and Formula Rearrangement

Duration: 1 Hour

Grade Level: Grade 9 Foundation

Learning Outcomes

By the end of this lesson you should be able to:

Understand and use the inequality symbols.
Solve simple linear inequalities.
Represent inequalities on a number line.
Rearrange simple formulae.
Make any variable the subject of a formula.

Part A – Study Guide

1. What is an Inequality?

An inequality compares two quantities that are not necessarily equal.

Symbols

Symbol	Meaning	Example
=	Equal to	(x = 5)
<	Less than	(x < 5)
>	Greater than	(x > 5)
≤	Less than or equal to	(x \leq 5)
≥	Greater than or equal to	(x \geq 5)

2. Solving Inequalities

The process is almost the same as solving equations.

Example 1

Solve

$x+4>10$

Subtract 4 from both sides.

$x>6$

Example 2

Solve

$x-7\le9$

Add 7.

$x\le16$

Example 3

Solve

$3x<21$

Divide by 3.

$x<7$

Important Rule

When multiplying or dividing by a negative number, the inequality sign reverses direction.

Example

Solve

$-2x>8$

Divide by -2.

Remember to reverse the sign.

$x<-4$

3. Number Line Representation

Example

$x>4$

Draw an open circle at 4 and shade to the right.

Example

$x\le4$

Draw a closed circle at 4 and shade to the left.

(Sketch these in your workbook.)

Formula Rearrangement

Formulae are used throughout Grade 10 and 11.

Sometimes you need to make a different variable the subject.

Example 1

Given

$y=x+4$

Make $x$ the subject.

Subtract 4.

$x=y-4$

Example 2

Given

$A=l\times w$

Make $l$ the subject.

Divide both sides by ( $w$ ).

$l=\frac{A}{w}$

Example 3

Given

$P=2l+2w$

Make ( $l$ ) the subject.

Subtract ( $2w$ ):

$P-2w=2l$

Divide by 2:

$l=\frac{P-2w}{2}$

Common Mistakes

Mistake 1

Forgetting to reverse the inequality sign when dividing by a negative.

Example

$-5x<20$

Correct answer:

$x>-4$

Mistake 2

Only moving one term in a formula.

Remember:

Treat formula rearrangement exactly like solving an equation.

Part B – Worked Examples

Example 1

Solve

$x+12>18$

Subtract 12.

$x>6$

Example 2

Solve

$4x\le20$

Divide by 4.

$x\le5$

Example 3

Solve

$-3x<15$

Divide by -3.

Reverse sign.

$x>-5$

Example 4

Make ( $a$ ) the subject.

$b=a+8$

Subtract 8.

$a=b-8$

Example 5

Make ( $w$ ) the subject.

$A=lw$

Divide by ( $l$ ).

$w=\frac{A}{l}$

Example 6

Make ( $r$ ) the subject.

$C=2\pi r$

Divide by ( $2\pi$ ).

$r=\frac{C}{2\pi}$

Part C – Practice Questions

Section A – Solve the Inequalities

x+5>12

x-9<4

2x>18

5x\le35

3x+6>18

4x-8\le20

7+x\ge18

9-x>3

-2x<10

-5x\ge25

Section B – Draw on a Number Line

For each inequality, draw a number line and show the solution.

x>2

x\le7

x<-1

x\ge5

x\le0

Section C – Rearranging Formulae

Make the variable shown in bold the subject.

y=x+9

Make $x$ the subject.

A=lw

Make $l$ the subject.

V=lwh

Make $h$ the subject.

C=2\pi r

Make $r$ the subject.

P=2l+2w

Make $w$ the subject.

F=ma

Make $m$ the subject.

I=\frac{V}{R}

Make $V$ the subject.

d=vt

Make $t$ the subject.

A=\frac12 bh

Make $h$ the subject.

M=\frac{x+y}{2}

Make $x$ the subject.

Part D – Mixed Questions

Solve

5x-10>20

Solve

4x+12\le36

Make ( $b$ ) the subject.

A=\frac12 bh

A cinema allows entry only to learners aged 13 years or older.

Write an inequality for the learner's age ( $a$ ).

A learner must score at least 50% to pass.

If the learner's percentage is ( $p$ ), write an inequality.

Challenge Questions

Solve

3x-5>16

Solve

-4x+8\le24

The perimeter of a rectangle is

P=2l+2w

Make ( $w$ ) the subject.

The formula for simple interest is

I=Prt

Make ( $r$ ) the subject.

A plumber charges a call-out fee of R250 plus R180 per hour.

Write a formula for the total cost ( $C$ ).

If a customer only has R970, what inequality can you write to find the maximum number of hours ( $h$ ) the plumber can work?

Answers

Section A

$(x>7)$
$(x<13)$
$(x>9)$

4. $(x\le7)$

$(x>4)$
$(x\le7)$
$(x\ge11)$
$(x<6)$
$(x>-5)$
$(x\le-5)$

Section B

Learner should draw:

Open circle at 2, shade right.
Closed circle at 7, shade left.
Open circle at -1, shade left.
Closed circle at 5, shade right.
Closed circle at 0, shade left.

Section C

$(x=y-9)$
$(l=\frac{A}{w})$
$(h=\frac{V}{lw})$
$(r=\frac{C}{2\pi})$
$(w=\frac{P-2l}{2})$
$(m=\frac{F}{a})$
$(V=IR)$
$(t=\frac{d}{v})$
$(h=\frac{2A}{b})$
$(x=2M-y)$

Section D

5x>30

x>6

4x\le24

x\le6

b=\frac{2A}{h}

a\ge13

p\ge50

Challenge

3x>21

x>7

-4x\le16

Divide by -4 and reverse the sign:

x\ge-4

w=\frac{P-2l}{2}

r=\frac{I}{Pt}

Formula:

C=250+180h

With a budget of R970:

250+180h\le970

Subtract 250:

180h\le720

Divide by 180:

h\le4

The plumber can work for 4 hours or less.

Parent's Notes

Today's lesson introduces skills that will be used throughout Grades 10 and 11, especially in:

Functions (rearranging equations).
Trigonometry (making a variable the subject).
Physics and Physical Sciences (formula manipulation).
Finance (using and rearranging interest formulas).

Encourage your student to write every step clearly. Most mistakes happen when learners try to do too much in their heads.

Looking Ahead – Week 2, Day 4

Tomorrow we'll cover Ratio, Proportion and Rates, including:

Simplifying ratios.
Dividing quantities in a given ratio.
Direct proportion.
Inverse proportion (introduction).
Unit rates and speed calculations.
Real-life word problems similar to those found in CAPS assessments.