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.