Example 1: Solve the inequality.
2  7 x / 5 > x + 3
Solution to example 1
multiply both sides by 5 (the LCD)
(2  7 x / 5)5 > (x + 3)5
10  7 x > 5 x + 15
subtract 10 from both sides.
 7 x > 5 x + 5
add 5 x to both sides.
2 x > 5
divide both sides by 2 and reverse inequality.
x < 5 / 2
conclusion: The solution set to the above inequality consists of all real numbers that are less than 5/2.
Example 2: Solve the polynomial inequality.
4 x^{ 2} > 4 x  8
Solution to example 2
Add  4 x + 8 to both sides to make one side zero
4 x^{ 2}  4 x + 8 > 0
Divide by the common factor  4 and reverse the inequality.
x^{ 2} + x  2 < 0
Factor the expression on the left.
(x  1)(x + 2) < 0
make table of signs of x  1, x + 2 and their product.
x inf 2 1 +inf
x  1     +
x + 2   +  +
(x  1)(x + 2) +    +
conclusion: The solutions of (x  1)(x + 2) < 0 are the values of x for which the resulting sign is negative.Thus, the solution set of the given inequality is the interval(2 , 1).
Example 3: Solve the rational inequality.
1 /(x + 4)  2 / (x  3) > = 0
Solution to example 3
Add the two rational expressions that are on the left
(x  11) / [ (x + 4)(x  3) ] > = 0
make table of signs of x  11, x + 4, x  3 and the whole expression on the left of the inequality.
x inf 11 4 3 +inf
x  11 +      
x  3       +
x + 4     +  +
(x  11) / [(x + 4)(x  3)] +    +  
conclusion: The solutions of 1 /(x + 4)  2 / (x  3) > = 0 are the values of x for which (x  11) / [(x + 4)(x  3)] is positive or is equal to zero.Thus, the solution set of the given inequality is the intervals (inf , 11 ] U (4 , 3).
More references and links on how to Solve Equations, Systems of Equations and Inequalities.
