ALGEBRA H4

If │2x – 7 │= 37; Find  x

 

Solution

 

±(2x – 11 )= 37

 

2x -11= 37    OR   –(2x-11) = 37

 

2x – 11 = 37    OR   -2x+11=37

 

2x = 37 + 11    OR   -2x= 37-11

 

2x = 48     OR   -2x  =   26

2      2               -2        -2

 

x = 24  OR  x = -13

 

 

Hence x = 14 OR x = -3

 

TRY THIS………………………. 

 

 

If │2x – 15 │= 63; Find x 

SIMPLIFY G8

 

Express as a single fraction m - 3 + m+5

                                              8         2

solution

 

= m - 3 + m+5

    8          2

 

= m - 3 + 4(m+5)                [Since the LCM of 8 and 2 is 8]

            8         

 

= m - 3 + 4m+20

            8         

 

= m + 4m + 20 - 3

            8         

 

= 5m +  17

        6                       

 

= 5m +  17       answer

        6         

 

 

TRY THIS...................

 

Express as a single fraction 7e - 1 + e+1

                                              3          6

FACTORIZE G9

 

Factorize 121-25m²

 

Solution

 

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

 

121-25m² = 11²-5²m²       

 

            = 11² - (5m)²       

 

            = (11 - 5m)(11 + 5m)

 

Hence 121-25m² = (11 - 5m)(11 + 5m)  answer

 

 

TRY THIS…………….

 

 

 

Factorize 64 – 49y²

MID-POINT E1

Find a and b if the midpoint of a line from (a, 6) to (8, b) is

(60, 70)

 

Solution

 

x1=a, x2=8, y1=6, y2=b.

 

(60, 70)= (x1 + x2, y1 + y2)

                   2             2

 

(60, 70)= (a + 8,  6 + b)

                  2          2

 

Equating equal values of x;

 

60= a + 8

         2      

 

2 x 60= (a + 8)  x 2 ¹

                2 ₁     

 

120 = a + 8

 

120-8 = a

 

a = 112.

 

Equating equal values of y;

 

70 = 6 + b

        2

 

2 x 70= (6 + b)  x 2 ¹  multiplying by 2 both sides.

                2 ₁     

 

140 = 6 + b

 

140 – 6 = b

 

b = 134

 

Hence a = 112 and b=134

 

TRY THIS……………………..

 

Find e and f if the midpoint of a line from (e, 19) to (13, f) is (10, 20)

 

 

GEOMETRICAL PROGRESSION A3

 

Find the sum of the 1st eight terms of the geometrical progression 4+16+64+…….

 

Solution

 

G₁ = 4, r=4, n=8

 

S_n = G₁(rⁿ-1)/ r-1

             

 

S₈ = 4(4⁸-1)/4-1

             

 

S₈ = 4(4⁸-1)/3 

             

 

S₈ = 4(262144-1)

             3

 

S₈ = 4(262,143)

             3

 

 

S₈ = 349,524

            

 

 

Hence the sum of the 1st eight terms is 349,524.

 

TRY THIS……………..

 

 

Find the sum of the 1st eight terms of the geometrical progression 3+12+48+…….

LOGARITHMS H6

 

Evaluate Log10,000,000,000 -  log 0.0001 + log₃243

 

Solution

 

= Log10,000,000,000 -  log 0.0001 + log₃243

 

= Log₁₀10¹⁰ -  log₁₀ 10^-4 + log₃3^-5

 

= 10Log10 -  (-4log 10) + (-5log₃3)

 

= (10x1) - (-4x1) + (-5x1)

 

= 10 - (-4) + (-5)

 

= 9

 

Hence Log10,000,000,000 +  log 0.0001 + log₃243 = 9

 

TRY THIS……………..

 

 

Evaluate Log1,000,000,000  -   log 0.000001 + log₃9

ARITHMETIC PROGRESSION 1F

 

The 1st term of an A.P. is 10 and the common difference is 58. Find the 16th term.

 

Solution

 

A₁= 10, d = 58

 

A_n = A₁ + (n-1)d   [ formula for n terms ]

 

A₁₆ = A₁ + (16-1)d

 

A₁₆ = A₁ + 15d    [ formula for 16 terms ]

 

A₁₆ = 10 + (16 x 58)

 

A₁₆ = 10 + 928

 

A₁₆ = 938

 

 

Hence the 16th term is 938.

 

TRY THIS……………..

 

 

The 1st term of an A.P. is 23 and the common difference is 58. Find the 22nd term.

FUNCTIONS 3G

 

Given that F(x) = 13x  -  55. Find F(8)

 

Solution

 

F(x) = 13x  - 55

 

F(8) = 13(8)  -  55

 

F(8) = 104  -  55

 

F(8) = 49

 

 

Hence  F(8) = 49

 

TRY THIS……………..

 

Given that F(x) = 55x  -  20. Find F(3)

SETS H21

 

In Ntoma district the number of people who speak Kiswahili or Lingala is 410. 200 of them speak Kiswahili and 320 of them speak Lingala. How many speak both languages?

 

solution

 

In most cases, OR stands for union whereas AND/BOTH, stands for intersection.

 

Let Kiswahili=n(K), Lingala= n(L).

 

n(K)= 200 ,

n(L)= 320,

n(KuL) = 410,

n(KnL)=?

 

 

 

n(KuL) = n(K) + n(L) - n(KnL)

 

 

410  =  200 + 320 – n(KnL)

 

 

410  =  520 – n(KnL)

 

 

n(KnL) =  520 – 410

 

 

n(KnL) =  110

 

Hence n(KnL)=110 answer

 

TRY THIS……………..

In Bibanja district the number of people who speak Spanish or German is 210. 170 of them speak Spanish and 180 of them speak German. How many speak both languages?