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Mathematics

Emmanuel Iwara(Tutor)
01-03-2016 12:40:00 +0000

Linear Inequality is most fundamental in the world of mathematics often tested by Jamb and other exams.

This session has ended for today

In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality: < is less than. > is greater than. ≤ less than or equal to.≥ greater than or equal to.

Solving inequalities does not create any problem, as the approach, manner and steps are the same as in linear equations.

Example 1 .Solve the inequality 4x + 12 <2

Solution

4x < 2 - 12(collecting like terms)

4x <-10

x < -10/4

x< -5/2

Example 2. Solve the Inequality

(3x + 4)/ x ≥ 3x/(x + 2)

Solution

Multiplying both sides by x(x+2)

x(x + 2)(3x + 4)/ x ≥ x(x+2)(3x)/ (x + 2)

= (x + 2)(3x + 4)≥ x(3x)

= [3x2 + 4x + 6x + 8) ≥ 3x2

Collecting like terms

3x2- 3x2 + 4x + 6x + 8 ≥ 0

= 10x + 8 ≥ 0

x = -8/10

Example 3. Solve the inequality

(2x2 + x ≥ x2 - 4x -6)

Solution

Collecting like terms we have

x2 + 5x + 6 ≥ 0

Factorizing

(x + 2)(x + 3) = 0

(x + 2 = 0 or x + 3 = 0)

(x = -2 or x = -3)

Example 4.Solve the inequality

6 < |x + 2|< 8

Solution

Case1
6 <|x+2|<8

x + 2 >6

x > 4

Case 2
(x + 2)< -6

x<-8

Thus

x>4 or x<-8

Also
|x + 2|<8

Case 1
x + 2 > 8

x > 6

and
Case 2
x + 2 > -8

x > -10

Written as -10<x < 6

Hence the solution to the problem above = (-10< x < -8 as well as 4 < x < 6)

Example 5. Example 5. Solve the rational inequality
(x2 - 3x + 2) / (x2 -9)≥ 0

Solution

(x2 - 3x + 2) / (x2 -9)≥ 0 = (x-1)(x-2)/(x-3)(x+3) when factorize

This rational function will be zero when the numerator is zero.

Thus
(x-1)(x-2) = 0

[(x-1) = 0 or (x-2) = 0]

[x = 1 or x = 2]

The rational function will be undefined wherever the denominator is zero.

Thus
(x -3)(x+3) = 0

(x-3) = 0 or (x+3) = 0

[x= 3 or x = -3]

Hence the solution is

x≤ -3; 1≤ x ≤ 2; x≥ 3.

Example 6.Solve |x-3| ≤6

Solution

Case 1.

x-3 ≤ 6
x≤ 6 +3

x≤ 9

and
x-3≥-6

x ≥-6 +3

x≥ -3

Hence solution is
x≥ -3 and x≤ 9

Now try and see if you can solve the following Past Jamb Questions

1. Determine the real numbers x satisfying x(x +x)>2.

2.List the Integers satisfying the inequality -2≤2x-6≥4

3. Find the range of values x which satisfy the inequality x/2 + x/3 + x/4 < 1

4.Solve the inequality (x-1)(x+2)/(x-4) ≤ 0

5.Solve |x+3| ≤ 4

• Emmanuel Blessing

dnt understand example 2. wrz d x(x+2) comin frm

0 01-03-2016 14:09:00 +0000

• Emmanuel Iwara(Tutor)

Hello class inequality is one of the most interesting topic one need to know as to advance in the world of mathematics. example 5 is less than x, implies that 5 is not inclusive but that the numbers are 1,2 3,and 4.

0 01-03-2016 14:15:00 +0000

• Emmanuel Blessing

5. x is greater or equal to -1 and x is lesser or equal to 9

0 01-03-2016 14:16:00 +0000

• Emmanuel Iwara(Tutor)

@Emmanuel Blessing x(x+2) is the L.C.M of the algebraic fraction (3x+4)/x greater than or equal to 3x/(x+2).

0 01-03-2016 14:21:00 +0000

• Emmanuel Blessing 0 01-03-2016 14:21:00 +0000

• Omotayo Olalekan

no.1 = x > 2/x+x

0 01-03-2016 14:24:00 +0000

• Emmanuel Blessing

plz @omotayo can u show me hw u got to dat answer

0 01-03-2016 14:27:00 +0000

• Emmanuel Iwara(Tutor)

@Emmanuel Blessing thanks for giving a try ,but want you to check your working solution to no.4. Stay close the solutions to the questions will be post after the section is over.

0 01-03-2016 14:31:00 +0000

• Omotayo Olalekan

no.4 x = 1 or x = -2

0 01-03-2016 14:32:00 +0000

• Emmanuel Iwara(Tutor)

@Omotayo thanks, but the question demand the real values of x

0 01-03-2016 14:33:00 +0000