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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