Understanding Engineering Mathematics

An inequality in whichf(x)is a linear (quadratic) function is called alinear (quadratic)
inequality, for example:

ax+b> 0 linear inequality

ax^2 +bx+c> 0 quadratic inequality

Example

If
1 < 3 x+ 2 < 4

then subtracting 2 from each term gives

− 1 − 2 =− 3 < 3 x< 4 − 2 = 2

So
− 3 < 3 x< 2

and so, dividing by the positive quantity 3, we obtain

− 1 <x<^23

Example

Find the range of values ofxfor which 2x^2 − 3 x− 2 >0.
It helps to factorise if we can, or find the roots of the quadratic:

(x− 2 )( 2 x+ 1 )> 0

This will be true if either:

x− 2 >0and2x+ 1 > 0 , i.e.x> 2

or if
x− 2 <0and2x+ 1 < 0 , i.e.x<−^12

So 2x^2 − 3 x− 2 >0forx<−^12 andx>2. You can also see this from the graph of the
function (➤RE (Reinforcement exercise) 3.3.2B), and graphical considerations often help
in dealing with inequalities generally. However, it is important that you gain practice in
the manipulation of inequalities, and the associated algebra.
Care is needed for inequalities involving the modulus. For example, suppose

|x− 3 |< 2

Then this means that

− 2 <(x− 3 )< 2

or − 2 + 3 = 1 <x< 2 + 3 = 5

so 1 <x< 5