Inequalities they are algebraic expressions armed with a inequality. They are very similar to the equations, especially with regard to the method of resolution and the way they are presented. What makes them different, among other things, is that the equations have a equality, and the inequalities, a inequality.
Equation x Inequation
The differences between equations and inequalities focus on the results, their analysis and quantity. To notice this difference, just follow the resolution of a problem that involves an equation and another that involves an inequality:
Equation: A young woman receives a salary of R$ 1200.00 at her job and wants to buy a car, which costs R$ 3200.00 in cash. Knowing that this young woman's expenses are approximately R$ 400.00 a month and that she is able to save the rest of the money without any problems, how long will it take for her to buy the car?
1200x - 400x = 32000
800x = 32000
x = 32000
800
x = 40
She will buy the car in 3 years and 4 months.
inequality: In an English school, a monthly fee of R$ 240.00 and a registration fee of R$ 100.00 are charged. What is the maximum number of months that a student who has R$2000.00 can attend?
100 + 240x < 2000
240x < 2000 – 100
240x < 1900
x < 1900
240
x < 7.92
The maximum number of months that this student can attend school is 7, as x is less than 7.92.
In this inequality, the result is accurate because we are looking for a “largest possible number”. However, inequalities normally do not have unique results. The results of inequalities are numerical sets and, most of the time, have infinite results.
When we look for the result of an equation, we look for a number that represents the accuracy of a situation. When we look for the result of an inequality, we are looking for a set of numbers that satisfies a certain sentence.
Inequality
THE inequality receives this name because it does not represent an equality. The symbols used are: , ≤ and ≥, which respectively mean: less, greater, less than or equal, greater or equal. To exemplify the use of these signs, note:
x > 2
This is the result of a inequality any and means that any number greater than 2 can be considered a correct answer. However, note that 2 is not greater than 2, so 2 itself does not satisfy the inequality.
x ≤ 6, with natural x
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Natural numbers are just non-negative integers. Therefore, the solutions for this inequality can also be written in list:
0, 1, 2, 3, 4, 5 and 6
This time, the number 6 is part of the list of solutions, due to the symbol “less than or equal”.
Inequality properties
Once aware of the above aspects, it is possible to think of solving methods for inequalities. Thanks to its similarity to equations, calculations must be performed in the same way. The only difference is in the inequality that will be put in place of equality. Because of this difference, the inequalities have some properties that need to be noted. Look:
Property 1 – Adding the same number to the two members of an inequality does not change the meaning of the inequality;
Property 2 – Subtracting the same number from the two members of an inequality does not change the meaning of the inequality.
Given the following inequality, note the solution:
15x - 9 < 5x +11
15x - 9 + 9 < 5x +11 + 9
15x < 5x + 20
15x – 5x < 5x + 20 – 5x
10x < 20
Property 3 – Multiply the two members of one inequality by a positive number does not change the direction of the inequality. Note the continuation of the solution to the above inequality, which will be multiplied by the positive number 1/10.
1 · 10x < 20 · 1
10 10
x < 2
This procedure is equivalent to “passing the 10 to the second member by dividing, since he is multiplying on the first”. Thus, this property is also valid as follows:
"Passing to the other member a positive number that is dividing or multiplying does not change the meaning of the inequality.”
Property 4 – Multiply the two members of one inequality by a negative number reverses the sign of inequality.
Thus, in cases where the inequalities need to be multiplied by – 1, this property must apply. For example:
4x - 9 > 12x + 23
4x – 12x > 23 + 9
– 8x > 32
Note that, in this step, the inequality must be multiplied by – 1. By property 4, we must invert the sign of inequality to get:
– 8x > 32 (– 1)
8x < – 32
x < – 32
8
x < – 4
By Luiz Paulo Moreira
Graduated in Mathematics
