Tutorial on Inequalities
This is a tutorial on solving polynomial and rational inequalities. Examples with step by step detailed solutions are presented
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.
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