LINEAR INEQUALITIES

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LINEAR INEQUALITIES IN ONE VARIABLE

Most of the rules for solving a linear inequalities in one variable are similar to those for solving a linear equation in one variable with exception of the rules on  multiplication and division by negative number which reverses the sense of the inequality

EXAMPLE : Find the solution set of each of the following inequalities and represent them graphically

(a) 2x – 3 < x + 7 (b) 3x  + 4 > 1 – 2x

Solution

(a)                  2x – 3 < x + 7

Adding 3 to both sides

2x < x + 10

Subtracting x from both sides

X < 10

(a)                  3x + 4 > 1 – 2x

Subtracting 4 from both sides

3x > – 3 – 2x

Adding 2x to both sides

5x >  -3

Dividing both sides by 5

x > -3/5

QUADRATIC INEQUALITIES IN ONE VARIABLE

To find the solution sets, of the quadratic inequalities of the form, ax² + bx + c ≥ 0 or ax² + bx + c ≤0. Note the following

1) If a>0 and b>0    then          a.b>0

or  a<0 and b<0     then          a.b>0

2) If a<0 and b>0    then         a.b< 0

Or a>0 and b<0    then         a.b< 0

Worked examples

1) Find the solution set of   x² + x – 6 > 0

Solution

x² + x – 6 > 0

(x – 2)( x + 3} > 0

x – 2> 0  or  x + 3<0

x >2   or      x < -3

x – 2 < 0    or  x +3>0

x < 2    or   x > -3

-3 < x < 2

 

-3 < x < 2

2)  Show graphically the solution Set of the inequality x² + 3x – 4 ≤ 0

Solution

X² + 3x – 4 ≤ 0

X² + 3x – 4 = 0

. (x – 1)(x + 4) ≤ 0

x – 1 ≤ 0   or   x + 4 ≥ 0

.x ≤ 1 or x ≥ -4

X – 1 ≥ 0   or x + 4 ≤ 0

X ≥ 1 or x ≤ -4

X ≤ 1 or x ≥ -4

– 4 ≤ x ≤ 1

Evaluation

Find the solution set of the inequalities

a) x² + 5x – 14 < 0

b) 2 – 3x – 9x²> 0

c) 1 – x² ≤ 0

Quadratic Inequality curve

We recall that th graph of  f(x) = ax^² + bx + c  is a parabola if D ≥ 0, the parabola crosses the axis at two distinct points, this fact can be used to solve the inequality ax² + bx + c ≥ 0 or ax² + bx + c ≤ 0

Worked examples

1) Determine the solution set of the inequality x² – x – 10 < 2

X² – x – 10 – 2 < 0

X² – x – 12 <  0

(x + 3)(x – 4)  < 0

x + 3 < 0 or x – 4 > 0

x < -3 or x > 4

x + 3 > 0 or x – 4 < 0

x > -3  or  x < 4

Using Parabolic curves  

Coordination of points at which the curve cuts the axis (x + 3)(x – 4) = 0

X = -3 , x = 4

2) Find the solution of the inequality x² – 2x – 3 ≥ 0

Solution

x² – 2x – 3 ≥ 0

(x + 1)(x – 3) ≥ 0

x + 1 ≥ 0 or x – 3 ≥ 0

x ≥ -1 or x ≥ 3

(x + 1) ≤ 0 or (x – 3) ≤ 0

x ≤ -1 or x ≤ 3

Solution set  -1  ≤  x  ≤  3

b)  9 – x² ≥ 0

3² – x² ≥ 0

(3 – x)(3 + x)  ≥  0

3 – x ≥ 0   or  3 + x ≥ 0

– x ≥ -3 or x ≥ -3

x ≤ 3  or  x  ≥ -3

(3 – x) ≤ 0 or (3 + x) ≤ 0

-x  ≤  -3  or  x  ≤  – 3

x ≥ 3 or x ≤ – 3

Solution set  -3  ≤  x  ≤  3

ABSOLUTE VALUES

If a number x is positive or negative the absolute value of x is denoted as │x│. The absolute value of a number is the magnitude of the number regardless of the sign.

Worked examples

1) │2x – 3│≥ 4

2x – 3 ≥ 4

2x ≥ 4 + 3

2x ≥ 7

.x ≥ 7/2

.x ≥ 3½

OR

​

– (2x – 3) ≥ 4

-2 x + 3 ≥ 4

– 2x ≥ 4 – 3

-2x ≥ 1

.x ≤ -½

Evaluation

Find the solution set of the inequality

a) │2x – 1│>3

b) │x – 3│ – │x – 1│< 0

c) │x – 3│ ≤│x – 2│

General Evaluation

1) Find the range of values of x for which 7x – 12 ≥ x²

2) For what values of x is 2x² – 11x + 12 positive?

3) Find the values of x satisfying: |3x – 2| ≥ 3|x – 1|

4)The 2^nd term of an exponential sequence is 9 while the 4^th term is 81. Find the common ratio, the first term and

the  sum of the first  five terms of the sequence.

5) Find the value of the constant k for which the equation 2x² + (k + 3)x + 2k = 0 has equal roots.

Reading Assignment : F/maths Project 1 pg 104 – 111

WEEKEND ASSIGNMENT

1) Find the range of x for which │2x – 1│> 3

(a) 1 < x < 3/2  (b) -3/2 < x < -1  (c) -3/2 < x < 1 (d) x > 3/2 and x < -1

2) Find the range of the value that satisfies the inequality x² + 3x – 18 < 0

(a) -3 < x < 6 (b)-3 > x <6 (c)-6 >x >3 (d)-6 >x < 3 (e)-6 < x <3

3) Find the range of values of x for which 2x² – 5x + 2 ≥ 0

(a) -2<x<-½ (b) ½ <x<2 (c) x < -½ or x ≥ -2 (d) x ≤½ or x ≥ 2

4) Find the range of values of y which satisfies the inequality 2y – 1 < 3 and 2 – y ≤ 5

(a) – 3 ≤ y ≤ 1  (b) – 2 ≤y ≤ 3 (c) -3≤ y ≤ 4 (d) -3 ≤ y ≤ 2

5) Find the range of values of x  for which 1/x + 3 < 2x is satisfy

(a) – 3 < x < 5/2  (b) x < -3 and x > -5/2  (c) x < 1 and x < ½

THEORY

1) Find the range of values of x for which 1 < x² – x + 1 < 7

2) Find the values of x satisfying |x – 5| – |x – 3| ≥ 0

3) Given that a and b are two real numbers, show that a² + b² ≥ 2ab.

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