Linear Inequalities

CONTENT

• Linear & Analytical Solutions of Linear Inequalities in One Variable
• Quadratic Inequalities in One Variables
• Absolute Values

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

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

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

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.

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.