LemonmathSelf-study sheet

Grade 9 · Lesson 2.2

Inequalities and their properties

Name: Date: Time spent: min

Goals

Read and write inequalities with the signs $<$ , $>$ , $\le$ , $\ge$ .
Use the transitive property and the rules for addition and multiplication to compare two expressions.
Prove some simple inequalities.

Remember

Inequalities: $a<b$ , $a>b$ , $a\le b$ , $a\ge b$ .
Transitive: $a<b$ , $b<c\Rightarrow a<c$ .
Adding (subtracting) the same number on both sides: the direction stays the same.
Multiplying (dividing) by a positive number: keep the direction; by a negative number: reverse it.
Proofs: reduce to $(\dots)^2\ge0$ .

Exercises

Exercise 1

Write as inequalities: a) the number of students $n$ in a class is at most $45$ ; b) the ticket price $p$ (thousand đồng) is at least $50$ ; c) the temperature $T$ ( $^\circ$ C) is below $0$ .

Exercise 2

Is each statement true or false? a) $-5<-3$ ; b) $2\cdot(-3)\ge-6$ ; c) $(-2)^2<3$ ; d) $7-10\le-4$ .

Exercise 3

Suppose $a<b$ . Compare: a) $a+5$ and $b+5$ ; b) $-3a$ and $-3b$ ; c) $2a-1$ and $2b-1$ .

Exercise 4

Given that $-2a+3<-2b+3$ , compare $a$ and $b$ .

Exercise 5

Prove that if $a>2$ , then $3a-1>5$ .

Exercise 6

A lift can carry at most $600$ kg. Mr Tư, who weighs $70$ kg, wants to take some bags of cement, each of $50$ kg. Write an inequality for the load with $10$ bags, and say whether he can take $10$ bags or $11$ bags.

Exercise 7 Review

What are the solutions of the equation $(x-1)(x+2)=0$ ? Which of them satisfies $x>0$ ?

Exercise 8

Suppose $a>b>0$ .

a) Prove that $a^2>ab$ .

b) Prove that $ab>b^2$ .

c) Deduce that $a^2>b^2$ .

Exercise 9 Challenge

Prove that $a^2+b^2+2\ge2(a+b)$ for all numbers $a$ , $b$ . When does equality hold?

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. "

x

does not exceed

5

" is written

x<5

.B.

x\le5

.C.

x\ge5

.D.

x>5

Question 2. Suppose

a<b

. Which statement is true?

-2a<-2b

.B.

a-3>b-3

.C.

3a<3b

.D.

-a<-b

Question 3. Suppose

a>b

. Compare

1-4a

and

1-4b

1-4a>1-4b

.B.

1-4a<1-4b

.C.

1-4a=1-4b

.D. They cannot be compared.

Question 4. Which statement is true for all numbers

a

b

A. If

a<b

, then

a^2<b^2

.B.

a^2+1>0

.C.

2a>a

.D. If

a<b

, then

-a<-b

Part II. True or false

Question 1. Suppose

a>b

a) $a+2>b+2$ .	T / F
b) $-5a>-5b$ .	T / F
c) $a-b>0$ .	T / F
d) $a^2>b^2$ .	T / F

Part III. Short answer

Question 1. How many integers

x

satisfy

-3<x\le2

Question 2. Find the least value of the expression

(x-3)^2+4

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: "at most" is $\le$ , "at least" is $\ge$ , "below" is $<$ .

Steps:

Decide the sign for each phrase.
Write the inequality.

Exercise 2.

What you need: work out each side, then compare; $\ge$ is also true when the two sides are equal.

Steps:

Work out each side.
Compare them on a number line.

Exercise 3.

What you need: adding keeps the direction; multiplying by a positive number keeps it, by a negative number reverses it.

Steps:

In b) multiply by $-3$ .
In c) multiply by $2$ , then add $-1$ .

Exercise 4.

What you need: add $-3$ to both sides, then divide by $-2$ (reverse the sign).

Steps:

Remove the $3$ from both sides.
Divide by $-2$ .

Exercise 5.

What you need: multiply by a positive number, then add the same number to both sides.

Steps:

Multiply both sides by $3$ .
Add $-1$ to both sides.

Exercise 6.

What you need: "at most $600$ " means the total mass is $\le600$ .

Steps:

Work out the total mass with $10$ bags and with $11$ bags.
Compare with $600$ .

Exercise 7.

What you need: equations in factorised form (previous lesson).

Steps:

Solve $x-1=0$ and $x+2=0$ .
Compare each solution with $0$ .

Exercise 8.

What you need: multiplying both sides by a positive number keeps the direction; the transitive property.

Steps:

a) Multiply $a>b$ by $a>0$ .
b) Multiply $a>b$ by $b>0$ .
c) Use the transitive property.

Exercise 9.

What you need: the square of any number is non-negative.

Steps:

Move everything to the left-hand side.
Write the left-hand side as $(a-1)^2+(b-1)^2$ .

Self-check

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐

Redo the ones you missed after: ☐ 1 day ☐ 3 days ☐ 7 days

Scan for step-by-step solutions

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐