**New QMaths 11B CD-ROM**

### Extra material

## Order and inequations

The usual order of numbers is determined by their size. The signs , , and are used to show the **order**

of numbers or algebraic expressions. In this section we will properly deﬁne these signs and the rules for their use.

The rules for the use of the order signs can all be proven from these deﬁnitions. The following rules are more
correctly called **theorems**. This means that they can be proven from deﬁnitions and basic facts (**axioms**). All
of the following apply to less than () in the same way as they are stated for greater than ().

Linear inequations can be solved in a similar way to linear equations. The difference is that multiplication
or division by a negative number *reverses* the order sign.

Solutions may be graphed on a number line or shown using interval notation.

**Inequality operators**

The inequality operator (**greater than**) is deﬁned as:

*ab*⇔ (*a*−*b*) is a positive number.
The operator (**less than**) is deﬁned as:

*ab*⇔*ba*

For convenience, we deﬁne (**greater than or equal**) and (**less than or equal**) as:

*ab*⇔*ab* or *a*=*b*
*ab*⇔*ab* or *a*=*b*

It also follows from these deﬁnitions that:

*ab*⇔ (*a*−*b*) is a negative number.

**!**

**Order theorems**

**Trichotomy:** For any two numbers *x* and *y*, exactly one of *xy*, *x*=*y* or *xy* is true.

**Transitivity:** If *xy*AND*yz*, then *xz*.
**Operations:** For all numbers *w*, *x*, *y*, *z*:

■ *xy*⇒*x*+*zy*+*z* ■ *xy*⇒*x*−*zy*−*z*

■ *xy*AND*z* 0 ⇒*xzyz* ■ *xy*AND*z* 0 ⇒*xzyz*

■ *xy*AND*z* 0 ⇒ ■ *xy*AND*z* 0 ⇒

■ *xy*AND*wz*⇒*x*+*wy*+*z*

*x*
*z*

-- *y*

*z*

-- *x*

*z*

-- *y*

*z*

**--!**

Detailed Proof

**Graphical notation**

■ A closed circle ( ) means the boundary is included.

■ An open circle ( ) means the boundary is not included.

■ A straight line on or above the number line shows the section included.

**Interval notation**

■ Numbers in brackets are used to indicate an **interval** (part of the number line).

■ Square brackets [ ] indicate the boundary is included.

**!**

Show the following using both graphs and interval notation.

**a** −2 *x* 3 **b** 4 *x* 9

**c** *x* 5 **d** *x*−3

Solution

**a**

**b**

**c**

**d**

−4 −3 −2 −1 0 1 2 3 4 or [−2, 3] as interval notation

2 3 4 5 6 7 8 9 10 or [4, 9) as interval notation

or (5, …] or (5, ∞] as interval notation

2 3 4 5 6 7 8 9 10

or […, −3] or [−∞, −3] as interval notation

−7 −6 −5 −6 −3 −2 −1 0 1

Example 1

Show the solutions of:

**a** *y*= 2*x*+ 2 where −3 *x* 3 **b** 12*x*+ 3*y*− 24 = 0 where −5 *x* 5

Solution

**a** Write the equation. *y* = 2*x*+ 2

Find the value of *y* when *x*=−3. *y* = 2 ×−3 + 2 =−4

Find the value of *y* when *x*= 3. *y* = 2 × 3 + 2 = 8

State the solution. −4 *y* 8

Now graph the solution and write the interval.

**b** Write the equation. 12*x*+ 3*y*− 24 = 0

Rearrange. 3*y* =−12*x*+ 24

Divide by 3. *y* =−4*x*+ 8

Find the value of *y* when *x*=−5. *y* =−4 ×−5 + 8 = 28

Find the value of *y* when *x*= 5. *y* =−4 × 5 + 8 =−12

State the solution. −12 *y* 28

Now graph the solution and write the interval.

or [−4, 8]

−6 −4 −2 0 2 4 6 8 10

16 20 24 28

or (−12, 28]

−16 −12 −8 −4 0 4 8 12

Solve and show the solutions of:

**a** 5*x*+ 11 2*x*− 7 **b** 7 − 4*yy*+ 4

**c** *p*− 3 4*p*+ 12 3*p*+ 10 **d** −

Solution

**a** Write the inequation. 5*x*+ 11 2*x*− 7

Collect variables (LHS) and constants (RHS). 5*x*− 2*x*−7 − 11

Simplify. 3*x*−18

Divide both sides by 3. *x*−6

Now graph the solution and write the interval.

**b** Write the inequation. 7 − 4*yy*+ 4

Collect variables and constants. −4*y*−*y* 4 − 7

Simplify. −5*y*−3

Divide both sides by −5. Remember, reverse the

sign when dividing by a negative. *y*

Now graph the solution and write the interval.

**c** Write the inequation.

This is solved as two separate problems.

*p*− 3 4*p*+ 12 3*p*+ 10

Write the ﬁrst inequation. *p*− 3 4*p*+ 12

Collect variables and constants. *p*− 4*p* 12 + 3

Simplify. −3*p* 15

Divide by −3 and reverse the sign. *p*−5

Write the second inequation. _{4}_{p}_{+}_{ 12}_{}_{ 3}_{p}_{+}_{ 10}

Collect variables and constants. 4*p*− 3*p* 10 − 12

Simplify. *p*−2

The answers can now be put together.

Now graph the solution and write the interval.

−5*p*−2

**d** Write the inequation. −

Multiply by the lowest common

denominator, 10. 10 − 10 10

Cancel where possible. 2(7*d*− 8) − (2*d*+ 7) 5(5*d*− 2)

Expand the brackets. 14*d*− 16 − 2*d*− 7 25*d*− 10

Simplify. 12*d*− 23 25*d*− 10

Collect variables and constants. 12*d*− 25*d* −10 + 23

Simplify. −13*d* 13

Divide by −13 and reverse the sign. *d* −1

7*d* –8
5

--- 2*d*+7
10

--- 5*d* –2
2

---or [−∞, −6]

−9 −8 −7 −6 −5 −4 −3 −2 −1

3 5

---or [−∞, )3 5

---−4 −3 −2 −1 0 3 1 2 3 4

5

---or (−5, −2]

−7 −6 −5 −4 −3 −2 −1 0 1

7*d*– 8
5

--- 2*d*+ 7
10

--- 5*d*–2
2

---7*d*– 8
5

--- 2*d*+ 7
10

**Exercise**

### Order and inequations

**1** Solve and show the following on a number line.

**a** 6*x*− 5 2*x*+ 31 **b** 2*r*+ 7 5*r*− 23

**c** *v*− 6 3*v*+ 12 **d** 5*e*− 3 *e*+ 9

**e** 3*m*+ 10 11*m*+ 34 **f** 5 −*ii*− 7

**g** 3*q*+ 5 5 − 3*q* **h** 6*k*+ 8 7*k*− 6

**i** 2*b*− 9 7*b*− 12 **j** 4*w*− 6 15 − 3*w*
**k** 19 − 3*a*−16 − 8*a* **l** 12*f*+ 6 3*f*− 27

**2** Solve and show the following on a number line.

**a** *y*= 3*x*+ 1 where −4 *x* 4 **b** *y*=−2*x*+ 5 where 0 *x* 8

**c** 2*y*= 6*x*+ 10 where −9 *x* 0 **d** *y*− 3*x*= 15 where −3 *x* 3

**e** 8*x*+ 2*y*− 12 = 0 where −5 *x* 4 **f** 9*x*+ 3*y*+ 21 = 0 where −7 *x* 3

**3** Solve and show the following as an interval and on a number line.

**a** *t*− 7 2*t*− 4 6 **b** 12 − 2*u* 5 − 3*u* 9 −*u*
**c** 3*n*+ 4 4*n*+ 9 1 **d** −19 3*g*− 7 *g*+ 1

**e** −2*x*− 1 7 −*x* 1 − 2*x* **f** 5*z*− 11 3*z*+ 5 4*z*+ 8

**g** 13*h*+ 4 9*h*− 8 15*h*+ 16 **h** 7 − 5*j* 9 − 6*j* 3

**4** Solve the following.

**a** − **b** + 0

**c** + *y*− 4 **d** − +

**e** − − **f** − −

**g** + 1 **h** − *k*−

**5** Prove each of the following theorems for real numbers *w*, *x*, *y*, *z*.

**a** *xy*AND*yz*⇒*xz* **b** *xy*⇒*x*−*zy*−*z*

**c** *xy*⇒*x*+*zy*+*z* **d** *xy*⇒*x*−*zy*−*z*
**e** *xy*AND*z* 0 ⇒*xzyz* **f** *xy*AND*z* 0 ⇒*xzyz*

**g** *xy*AND*z* 0 ⇒*xzyz* **h** *xy*AND*z* 0 ⇒

**i** *xy*AND*z* 0 ⇒ **j** *xy*AND*z* 0 ⇒

**k** *xy*AND*z* 0 ⇒

**l** *wx*AND*yz*⇒*w*+*yx*+*z* (*Hint:* Use transitivity after adding *y* to the ﬁrst relation
and *x* to the second relation.)

**m** *xy* 0 ⇒ 0 (*Hint:* Use an operation with *xy*.)

**n** For all values of *x*, *y* 0: *xy*⇔*x*2_{}* _{y}*2

_{(You must prove both ways.)}3

*x*–1

4

--- 2*x*+3
3

--- 4*x*– 9
12

--- 2*w*–1

7

--- 2*w*+9
3

---2*y* –5
3

--- *y*– 4
2

--- 3*p*–8

5

--- 2*p*+1
4

--- 3*p*–5
10
--- 1
4
*---t*
6

--- 3*t*–2
8

--- 4*t*+11
12

--- *t*–2
4

--- 3*v*+5

8
--- *v*

3

--- *v*+3
3

--- *v*+ 7
4

---2(*m*+5)
5

--- 3*m*+1
4

--- 2 5

--- 3*k*+1

8 --- 1

6

**---Exercise**

### Answers

**1 a** *x* 9 **b** *r* 10

**c** *v*−9 **d** *e* 3

**e** *m*−3 **f** *i* 6

**g** *q* 0 **h** *k* 14

**i** *b* **j** *w* 3

**k** *a*−7 **l** *f*−3

**2 a** −11 *y* 13 **b** −11 *y* 5

**c** −22 *y* 5 **d** 6 *y* 24

**e** −10 *y* 26 **f** −16 *y* 14

**3 a** −3 *t*5 [−3, 5] **b** −7 *u*−2 (−7, −2)

**c** −5 *n*−2 [−5, −2] **d** −4 *g*4 (−4, 4)

**e** −8 *x*−6 (−8, −6] **f** −3 *z*8 [−3, 8)

**g** −4 *h*−3 (−4, −3] **h** 1 *j*2 [1, 2)

**4 a** *x*−2 **b** *w*−3 **c** *y*−2 **d** *p*−8

**e** *t*−4 **f** *v* 33 **g** *m* 1 **h** *k*−17

6 7 8 9 10 12 13 14 15 7 8 9 10 11 12 13 14 15

−12 −11 −10 −9 −8 −7 −6 −5 −4 0 1 2 3 4 5 6 7 8

−6 −5 −4 −3 −2 −1 −2 −3 −4 3 4 5 6 7 8 9 10 11

−3 −2 −1 0 1 2 3 4 5 11 12 13 14 15 16 17 18 19

3 5

---−5 −2 −1 0 1 2

3 5

---0 1 2 3 4 5

3 4 5 6 7 8

2 3

---−6 −5 −4 −3 −2 −1

2 3

---−3

−10 −9 −8 −7 −6 −5 −6 −7 −8 −2 −3 −4

−12 −8 −4 0 4 8 12 16

13

−11

−12−10 −8 −6 −4 −2 0 2

5

−11

4 6

20

−30−20 −10 0 10 20 30 40

5

−22

4 8 12 16 20 24 28

6

50 60 32 36 40

−10 −5 0 5 10 15 20 25

26

30 −20−16−12 −8 −4 0 4

14

8 12 16

−8 −6 −4 −2 0 2 4 6 8 −8 −7 −6 −5 −4 −3 −2 −1 0

−8 −7 −6 −5 −4 −3 −2 −1 0 −8 −6 −4 −2 0 2 4 6 8

−10 −9 −8 −7 −6 −5 −4 −3 −2 −8 −6 −4 −2 0 2 4 6 8

−7 −6 −5 −4 −3 −2 −1 0 1 −4 −3 −2 −1 0 1 2 3 4