Inequalities
The inequalities section of QuickMath allows you to solve virtually whatsoever inequality or system of inequalities in a single variable. In nearly cases, you can find exact solutions. Fiftyfifty when this is not possible, QuickMath may exist able to give you approximate solutions to almost any level of accuracy you require. In addition, you lot can plot the regions satisfied past one or more than inequalities in two variables, seeing clearly where the intersections of those regions occur.
What are inequalities?
Inequalities consist of two or more than algebraic expressions joined by inequality symbols. The inequality symbols are :
<  less than 
>  greater than 
<=  less than or equal to 
>=  greater than or equal to 
!= or <>  not equal to 
Here are a few examples of inequalities :
two 10 – 9 > 0

x^{2}

 5x – ane  <> 5

x^{3}

Solve
The Solve command can exist used to solve either a single inequality for a single unknown from the basic solve page or to simultaneously solve a system of many inequalities in a unmarried unknown from the avantgarde solve page. The avantgarde control allows you lot to specify whether you want approximate numerical answers also every bit exact ones, and how many digits of accuracy (up to 16) you require.
Multiple inequalities in the avantgarde section are taken to be joined past AND. For example, the inequalities
2 x – 1 > 0
x^ii – v < 0
on two separate lines in the advanced department are read by QuickMath as
2 x – i > 0 AND ten^2 – 5 < 0
In other words, QuickMath will endeavour to notice solutions satisfying both inequalities at once.
Become to the Solve folio
Plot
The Plot command, from the Graphs section, will plot whatsoever inequality involving ii variables. In order to plot the region satisfied past a single inequality involving ten and y, go to the basic inequality plotting page, where you tin enter the inequality and specify the upper and lower limits on ten and y that you want the graph to exist plotted for. The avantgarde inequality plotting page allows you lot to plot the union or intersection of upward to 8 regions on the 1 graph. You have control over such things as whether or not to bear witness the axes, where the axes should exist located and what the aspect ratio of the plot should be. In addition, you lot have the option of showing each private region on its own.
Introduction to Inequalities
An equation says that two expressions are equal, while an inequality says that one expression is greater than, greater than or equal to, less than, or less than or equal to, some other. As with equations, a value of the variable for which the inequality is true is a solution of the inequality, and the ready of all such solutions is the solution gear up of the inequality. Two inequalities with the same solution set are equivalent inequalities. Inequalities are solved with the following properties of inequality.
PROPERTIES OF INEQUALITY
For real numbers a, b, and c:
(a)
(The same number may exist added to both sides of an inequality without changing the solution set.)
(b)
(Both sides of an inequality may exist multiplied past the same positive number without changing the solution gear up.)
(c)
(Both sides of an inequality may exist multiplied past the same negative number without changing the solution gear up, every bit long as the management of the inequality symbol is reversed.)
Replacing < with >, results in equivalent backdrop.
NOTE
Because division is defined in terms of multiplication, the word “multiplied” may be replaced by “divided” in parts (b) and (c) of the backdrop of inequality.
Pay conscientious attention to part (c): if both sides of an inequality are multiplied past a negative number, the direction of the inequality symbol must be reversed. For case, starting with the truthful statement – 3 < v and multiplying both sides by the positive number 2 gives
still a true statement. On the other hand, starting with – 3 < 5 and multiplying both sides by the negative number 2 gives a truthful issue only if the direction of the inequality symbol is reversed.
A similar situation exists when dividing both sides by a negative number. In summary, the following statement can be made.
When multiplying or dividing both sides of an inequality by a negative number, we must reverse the direction of the inequality symbol to obtain an equivalent inequality.
LINEAR INEQUALITIES
A linear inequality is defined in a manner similar to a linear equation.
A linear inequality in one variable is an inequality that can be written in the form
where a <> 0.
Example 1
Solve the inequality 3x + v > 7.
Use the properties of inequality. Calculation – 5 on both sides gives
Now multiply both sides by 1/3. (We could also separate by 3.) Since 1/3 < 0, opposite the direction of the inequality symbol.
The original inequality is satisfied by any existent number less than 4. The solution gear up can be written {xx < four}. A graph of the solution ready is shown in Effigy 2.6, where the parenthesis is used to show that 4 itself does not belong to the solution set up.
The set {x10 < 4}, the solution set for the inequality in Example one, is an example of an interval. A simplified notation, called interval annotation, is used for writing intervals. With this annotation, the interval in Instance i tin exist written as (oo, 4). The symbol oo is not a real number; it is used to prove that the interval includes all existent numbers less than four. The interval (oo, four) is an example of an open up interval, since the endpoint, 4, is not part of the interval. Examples of other sets written in interval notation are shown beneath. A foursquare subclass is used to show that a number is office of the graph, and a parenthesis is used to indicate that a number is not part of the graph. Whenever two real numbers a and b are used to write an interval in the chart that follows, information technology is assumed that a < b.
Example ii
Solve four – 3y < 7 + 2y. Write the solution in interval notation and graph the solution on a number line. Write the following series of equivalent inequalities.
In setbuilder note, the solution set is {yy>=3/5}, while in interval notation the solution gear up is (iii/5, oo). Run into Figure 2.seven for the graph of the solution set.
From now on, the solutions of all inequalities will be written with interval notation.
THREEPart INEQUALITIES
The inequality two < 5 + 3m < twenty in the next example says that 5 + 3m is between 2 and 20. This inequality can be solved using an extension of the backdrop of inequality given above, working with all 3 expressions at the same fourth dimension.
Solve 2 < 5 + 3m < 20.
Write equivalent inequalities equally follows.
The solution is graphed in Figure 2.8
QUADRATIC INEQUALITIES
The solution of quadratic inequalities depends on the solution of quadratic equations.
A quadratic inequality is an inequality that tin be written in the form
Nosotros will discuss quadratic inequalities in the next section.
Get to the Inequalities Plotting page
Solve the Following Inequality 38 4x 3 7 3x
Source: https://quickmath.com/pages/modules/inequalities/index.php