Introduction come Inequalities

Inequalities are offered to demonstrate relationships between numbers or expressions.

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Key Takeaways

Key PointsAn inequality explains a relationship between two different values.The notation a b way that a is strictly better than b.The id a leq b means that a is less than or same to b, when the notation a geq b way that a is greater than or same to b.Inequalities are specifically useful for solving troubles involving minimum or maximum possible values.Key Termsnumber line: A visual depiction of the set of actual numbers as a collection of points.inequality: A statement that of two amounts one is specifically much less than or higher than another.

In mathematics, inequalities are supplied to compare the relative size of values. They deserve to be supplied to compare integers, variables, and also various various other algebraic expressions. A description of different varieties of inequalities follows.

Strict Inequalities

A strict inequality is a relation that holds between two values as soon as they are different. In the same method that equations usage an equals sign, =, to present that 2 values are equal, inequalities use indications to display that two values are not equal and also to describe their relationship. The strict inequality symbols room .

Strict inequalities differ from the notation a eq b, which method that a is no equal to b. The eq prize does not say that one value is higher than the other or also that they deserve to be contrasted in size.

In the two varieties of strictly inequalities, a is not equal to b. To to compare the dimension of the values, there space two types of relations:

The notation a The notation a > b method that a is better than b.

The definition of these symbols can be easily remembered by noting that the “bigger” side of the inequality price (the open side) encounters the bigger number. The “smaller” side of the prize (the point) faces the smaller sized number.

The above relations have the right to be demonstrated on a number line. Recall that the worths on a number line rise as you move to the right. The following therefore represents the relation a is much less than b:

*

a b

a is come the ideal of b top top this number line.

In general, keep in mind that:

a a; because that example, 7 7.a > b is indistinguishable to b 6.

Other Inequalities

In comparison to strict inequalities, there room two varieties of inequality relations that room not strict:

The notation a leq b means the a is less than or equal to b (or, equivalently, “at most” b).The notation a geq b means that a is higher than or same to b (or, equivalently, “at least” b).

Inequalities through Variables

In enhancement to mirroring relationships between integers, inequalities can be supplied to present relationships between variables and integers.

For example, take into consideration x > 5. This would certainly be review as “x is greater than 5″ and indicates the the unknown variable x can be any kind of value higher than 5, though no 5 itself. Because that a visualization of this, check out the number heat below:

x > 5

Note the the circle above the number 5 is open, indicating that 5 is not contained in feasible values that x.

For another example, think about x leq 3. This would be read as “x is less than or equal to 3″ and also indicates the the unknown variable x can be 3 or any value less than 3. For a visualization of this, check out the number heat below:

x leq 3

Note that the circle above the number 3 is filled, indicating that 3 is had in feasible values the x.

Inequalities room demonstrated by color in an arrow over the appropriate variety of the number heat to suggest the feasible values that x. Keep in mind that an open up circle is used if the inequality is strict (i.e., because that inequalities using > or x - 7 > 122x + 4 leq 252x

Each of this represents the relationship between two different expressions.

One valuable application of inequalities such together these is in problems that involve preferably or minimum values.

Example 1

Jared has a boat with a maximum weight limit of 2,500 pounds. He wants to take as numerous of his girlfriend as feasible onto the boat, and he guesses the he and his friends weigh an median of 160 pounds. Exactly how many people can drive his boat at once?

This trouble can it is in modeled v the following inequality:

160n leq 2500

where n is the variety of people Jared can take top top the boat. To see why this is so, consider the left side of the inequality. It represents the total weight of n people weighing 160 pounds each. The inequality states that the total weight the Jared and his friends must be less than or equal to the maximum weight of 2,500, i beg your pardon is the boat’s weight limit.

There are procedures that deserve to be complied with to fix an inequality such as this one. For now, the is essential simply to know the an interpretation of together statements and cases in i beg your pardon they might be applicable.

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Rules for fixing Inequalities

Arithmetic operations can be provided to deal with inequalities because that all feasible values that a variable.