Linear Inequalities with One Variable

Here we talk about inequalities of the form ax>b or ax<b,ax>=b,ax<=b. If a>0 then inequality ax>b is equivalent to the inequality x>b/a, i.e. set of solutions of this inequality is interval (b/a,+oo). If a<0 then inequality ax>b is equivalent to the inequality x<b/a, therefore, set of solutions of this inequality is interval (-oo,b/a).

At last, if a=0 then inequality has form of 0*x>b, i.e. it doesn't have solution when b>=0 and is true for all x when b<0.

Example . Solve 2(x-3)+5(1-x)>=7(2x-5).

After a simplification we obtain:

2x-6+5-5x>=14x-35;

2x-5x-14x>=-35+6-5;

-17x>=-34.

Now, we divide inequality by -17 and change sign of inequality: x<=2.

Therefore, (-oo,2] is set of solutions of given equation.