INEQUALITIES 1

Find x if 5x – 29 ≤ x + 15 ≤ 6x.

Solution

5x – 29 ≤ x + 15 and  x + 15 ≤ 6x

5x – x ≤ 29+ 15 and  15 ≤ 6x - x

4x ≤ 44 (divide by 4 both sides) and  15 ≤ 5x (divide by 5 both sides)

x ≤ 11 and  3 ≤ x

x ≤ 11 and  x ≥ 3 answer

TRY THIS……..

Find x if 6x – 55 ≤ x + 15 ≤ 2x. 

LOGARITHMS 15

Evaluate Log100 +  log 0.001 - log 0.0001+ log 0.00001

Solution

= Log100 + log 0.001 -  log 0.0001+ log 0.00001

= Log10² +  log 10^-3 – log 10^-4 + log 10^-5

= 2Log10 + (-3 log 10) – (-4log 10) + (-5log 10)

= (2x1) + (-3x1) - (-8x1) + (-5 x 1)

= 2 + (-3) – (-4) -5

= 2 - 3 + 4 - 5

= 2 + 4 - 3 - 5

= 6 - 8

= 2  

Hence Log100 + log 0.001-  log 0.0001+ log 0.00001 = 2

TRY THIS……………………… 

Evaluate Log100 +  log 100,000 - log 0.0001+ log 0.00001

WORD PROBLEMS 3

The difference between two numbers is 21 and their sum is 683. Find the two numbers.

Solution

Let the two numbers be x and y.

x – y = 21 ----------- (i)

x +y = 683 ------------(ii)

---------------------------------------------

From (i)

 x – y = 21

x = 21+ y -------------- (iii)

-------------------------------------------

Substitute (iii) in (ii) above.

x +y = 683. ------------(ii)

(21+ y) +y = 683

21 + 2y = 683

2y = 683 - 21

2y = 662

¹2y = 662³³¹

₁2      2₁

y = 331

from (iii)

x = 21+ y

x = 21+ 331

x = 352

Hence the two numbers are 331 and 352.

TRY THIS…………………………………   

The difference between two numbers is 33 and their sum is 167. Find the two numbers.

WORD PROBLEMS 2

When 7 is subtracted from twice a certain number, the result is greater than 93. Find the number.

Solution

Let the number be w.

2w – 7 > 93

2w > 93 + 7

2w > 100

2w > 100

2        2

w > 50 answer

TRY THIS…………………………… 

When 30 is subtracted from thrice a certain number, the result is greater than 76. Find the number.

RATIOS 1

Decrease 87000sh by 3:10.

Solution

= 3 x 87000sh

   10

= 3 x 87000⁸⁷⁰⁰

   10₁

= 3 x 8700

= 26,100Tshs answer

TRY THIS………………………………………… 

Decrease 28,000sh by 7:20.

SIMPLIFY 4

Simplify 20 + 6(100n ÷ ⁿ/2) - 80

Solution

=20 + 6(100n ÷ ⁿ/2) - 80

=20 + 6(100n x ²/n) – 80

=20 + 6(100n¹ x ²/n₁) – 80  cancelling n

=20 + 6(100 x  2) – 80 

=20 + 6(200) – 80 

=20 + 1200 – 80 

=1220 – 80 

=1140

=1140answer

TRY THIS……..

Simplify 70 + 4(10n ÷ ⁿ/5) - 35

WORD PROBLEMS 1

Ana, Joy and Fred shared 2400 sweets such that Ana got 50 sweets less than Joy while Fred got three as much as what Ana got. Find those given to Fred.

Solution

The starting point is Joy. Let the sweets given to Joy be x. See the table below:

ANA	JOY	FRED
x-50	x	3(x-50)=3x-150

  

Then;

ANA + JOY + FRED = 2400

x-50 + x + 3x-150 =2400

x + x + 3x - 150 - 50 = 2400

5x - 200 = 2400

5x = 2400 + 200

5x = 2600

5x = 2600

5       5

x = 520

The sweets given to Fred

= 3x – 150

= 3(520) – 150

= 1560 – 150

=1410

Hence Fred got 1410 sweets.

TRY THIS……………………………………… 

Kemi, Roy and Isaya shared 246 sweets such that Kemi got 10 sweets less than Roy while Isaya got four as much as what Kemi got. Find those given to Isaya.   [132 answer]

FUNCTIONS 5

If F(x) = 4x + 13; Find F^-1(x).

Solution

HINT: F^-1(x) means inverse.

PROCEDURE:

Make x the subject and then interchange x and y variables.

Let y=F(x)

So,  y= 4x + 13

y – 13 = 4x

y – 13  = 4x

   4          4

y – 13   = x

   4

x  =   y – 13     after rearranging

           4

F^-1(x) = x – 13    after interchanging x and y variables.

                4

Hence, F^-1(x) = x – 13     

                              4

TRY THIS………………   

If F(x) = 7x + 20; Find F^-1(x).

SIMPLIFY 3

Simplify  8w(5w – 7 + w) + 12w².

solution

= 8w(5w – 7 + w) + 12w².

= (8w x 5w) – (8w x 7) + (8w x w) + 12w².

= [40w² – 56w + 8w² ]+ 12w².

= 40w² + 8w² + 12w² – 56w

= (48w² + 12w²)  – 56w

= 60w² – 56w

= 60w² – 56w answer

TRY THIS………..

Simplify 3a(7a+ 30 - a) +29a².

FACTORIZE 1

Factorize 81 - 225m²

Solution

We use difference of two squares a² – b² = (a - b)(a + b)

81 - 225m² = 9² - 15²m²       

                = 9² – (15m)²       

                = (9 - 15m)(9 + 15m)

Hence 81 - 225m² = (9 - 15m)(9 + 15m) answer

TRY THIS…………….

Factorize 49 – 121y²