Lesson Notes By Weeks and Term - Senior Secondary School 2

ALGEBRAIC FRACTIONS

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SUBJECT: MATHEMATICS

CLASS:  SS 2

DATE:

TERM: 2nd TERM

REFERENCE BOOKS

  • New General Mathematics SSS2 by M.F. Macraeetal.
  • Essential Mathematics SSS2 by A.J.S. Oluwasanmi. 

 

 
WEEK FIVE                                    DATE: ___________

TOPIC: ALGEBRAIC FRACTIONS

CONTENT

-Substitution in Fractions.

-Undefined Fractions.

 

SUBSTITUTION IN FRACTIONS 

Example 1

Given that x:y = 9:4, evaluate 8x-3yx- 34

If x:y = 9:4, then xy= 94

Divide numerator and denominator of 

8x-3yx- 34yby y. 

8x-3yx- 34y = 8xy- 3xy- 34

 

Substitute 94 for xy in the expression. 

Value of expression 

= 8 × 94- 394- 34 = 18-3 112= 15112

= 15 ÷ 32=15 × 23=10

 

Example 2

If x = 2a+33a-2 , express x-12x+1 in terms of a. 

Substitute 2a+33a-2 for x in the given expression. 

x -12x +1= 2a +33a -2- 12 × 2a +33a -2+ 1

Multiply the numerator and denominator by (3a – 2). 

 

x-12x+1= 2a +3- (3a-2)22a +3+ (3a -2)

    = 2a+3-3a+24a+6+3a -2

    = -a+57a+4 or 5 -a4 +7a

 

Example 3 

Solve the equation 13a -1= 2a+1- 38

The LCM of the denominators is 8(3a – 1)(a + 1).

To clear fractions, multiply the terms on both sides of the equation by 8(3a – 1)(a + 1).

If 13a -1= 2a+1- 38

Then 13a -1 ×83a-1(a+1)

    = 2a -1=83a-1a+1

    = -38 ×83a-1a+1

8(a + 1) = 16(3a – 1) – 3(3a – 1)(a + 1)

    8a + 8 = 48a – 16 – 3(3a2 + 2a – 1)

    8a + 8 = 48a – 16 – 9a2 – 6a + 3

8a + 8 – 48a + 16 + 9a2 + 6a – 3 = 0 

            9a2 – 34a + 21  = 0 

            (a – 3)(9a – 7) = 0

a = 3 or 9a = 7 

a = 3 or 7/9

 

Check: if a = 3,    13a-1= 19-1= 18

and        2a+ 1- 38= 24- 38= 12- 38= 18

if a = 79,    13a-1= 173-1= 134= 34

and        2a+1- 38= 2179- 38

            = 1816- 38

            = 98- 38= 34

EVALUATION

  1. if xy= 34, evaluate 2x-y2x+y .
  2. If x = a+32a- 1, express 2x+13x+1 in terms of a. 

 

UNDEFINED FRACTIONS 

If the denominator of a fraction has the value zero, the fraction will be undefined. If an expression contains an undefined fraction, the whole expression is undefined. 

 

Example 1 

Find the values of x for which the following frxactions are not defined. 

  1. 3x+2     b. 2x+133x-12

 

  1. 3x+2 is undefined when x + 2 = 0 

    if x + 2 = 0 

    then x = -2

    the fraction is not defined when x = -2.

 

  1. 2x+133x-12 is undefined when 3x – 12 = 0.

    If 3x – 12 = 0 

    Then 3x = 12

        x = 4

 

Example 2 

Find the values of x for which the expression 

ax- bx2+6x-7is not defined.

ax- bx2+6x-7= ax- bx-1(x+7)

The expression is not defined if any of the fractions has a denominator of 0.

axis undefined when x = 0.

(x – 1)(x + 7) = 0

If (x – 1)(x + 7) = 0

Then either (x – 1) = 0    or (x + 7) = 0

i.e. either x = 1 or x = -7

The expression is not defined 

When x = 0, 1 or -7

 

Example 3

  1. For what value(s) of x is the expression x2 +15x +50x-  5 not defined? 
  2. Find the value(s) of x for which the expression is zero. 

 

Solution 

  1. The expression is undefined when its denominator is zero, 

i.i. when x – 5 = 0

    x = 5 

 

  1. let x2 +15x +50x-  5 = 0 

multiply both sides by x – 5

x2 + 15x + 50 = 0

(x + 5)(x + 10) = 0

Either x + 5 = 0 or x + 10 = 0 

i.e. either x = -5 or x = -10 

The expression is zero when x = -5 or x = -10.

 

EVALUATION 

For what value(x) of x are the following expressions (i) undefined   (ii) equal to zero?

  1.   815+3x 2.  5b(1-2x)x

 

GENERAL EVALUATION/ REVISION QUESTIONS

  1. If x = 3m-53m+5, express x-1x+1 in terms of m.
  2. If X = 2a+33a-2, express X-12X+1 in terms of a. 
  3. If h = m+1m-1, Express 2h-12h+1 in terms of m.
  4. Solve the following.
  5. a)   3a = a – 2         b.)      5 – 2d = 2d ) 73+ 2e=e        d.) 2m+32m+5- m-1m-2=0

e.)3c+2- 22c-3= 17

 

WEEKEND ASSIGNMENT

Objectives

1.For what values of x is the expression7x2x+1(x-1) not defined? A.1, B. -1,-1   C. -1,1 D.    2,1   

2.For what values of x is the expression1x2-3x+2  not defined? A. 1,2  B. -1,2 C. -1,-2 D. 1,-2 

3.Solve3+xx  = 0      A. 1  B.  3  C.   -3   D.  -1   

  1. Simplify 32x-4+ 26-3x A. 5(2-x)2x-4(6-3x) B. 5(x-2)2x-4(6-3x) C. 5x+32x-4(6-3x)  D.   5x-32x-4(6-3x)   
  2. For what value of x is the expression 7x2x+1(x-1) equal to zero?  A. 0 B. 1 C. 2 D. 3

 

Theory

  1. a. For what value(s) of x is the expression 2x+11x2+x-20 not defined? 
  2. For what value(s) of x is the expression zero? 

 

  1. if a = 2m+12m-1, express 2a+12a-1 in terms of m. 

 

READING ASSIGNMENT

New General Mathematics SSS2, pages 195-201, exercise 17f and 17g.





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