math [1.11-01] the philosophy of inequalities

 


Inequalities are expressions where one side of the relational symbol is one of the following: (a) less than, (b) greater than, (c) less than or equal to, or (d) greater than or equal to.

A linear inequality dictates that two expressions are not equivalent.

Linear inequalities can have multiple solutions and their solutions can be graphed on a number line.

Consider the following linear inequality: [x - 1 3]. If we add [1] to both sides of the inequality, we get [x 4].

So, [x] is less than or equal to [4].

The solution set for [x] is less than or equal to [4] is (-, 4].

Notice the parenthesis [ ( ] on the left side of the solution set and the bracket [ ] ] on the right side.

The bracket indicates the possibility that the solution is equal to 4.

If the solution set were less than but not equal to [4], then there would be a parenthesis [ ) ] on the right side of the solution set rather than a bracket.

The number line graph of the solution set for [x] is less than or equal to [4] is given in the graph below:

Properties of Inequalities:

Addition Property of Inequality - For all real numbers [a], [b], and [c], the inequalities [a < b] and [a + c < b + c] are equivalent.

This is true for the greater than [>] sign, the less than or equal sign [] and the greater than or equal sign [].

Multiplication Property of Inequality - For all real numbers [a], [b], and [c], where [c 0], the inequalities [a < b] and [ac > bc] are equivalent if [c] is greater than zero.

This is true for the greater than [>] sign, the less than or equal sign [] and the greater than or equal sign [].

Procedure for solving a linear equation with one variable:

[1]. Simplify both sides of the inequality through combining like terms and removing parenthesis

[2]. Use the addition property of inequality to get the variable on one side

[3]. Use the multiplication property of inequality to create the form [x < y] or [x > y].

If and only if multiplying or dividing by a negative number, the inequality sign will be reversed.



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