


InequalitiesThe inequalities section of QuickMath allows you to solve virtually any inequality or system of inequalities in a single variable. In most cases, you can find exact solutions. Even when this is not possible, QuickMath may be able to give you approximate solutions to almost any level of accuracy you require. In addition, you can plot the regions satisfied by one or more inequalities in two variables, seeing clearly where the intersections of those regions occur. What are inequalities?Inequalities consist of two or more algebraic expressions joined by inequality symbols. The inequality symbols are :
Here are a few examples of inequalities :
SolveThe Solve command can be 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 single unknown from the advanced solve page. The advanced command allows you to specify whether you want approximate numerical answers as well as exact ones, and how many digits of accuracy (up to 16) you require. Multiple inequalities in the advanced section are taken to be joined by AND. For example, the inequalities 2 x  1 > 0x^2  5 < 0 on two separate lines in the advanced section are read by QuickMath as 2 x  1 > 0 AND x^2  5 < 0In other words, QuickMath will try to find solutions satisfying both inequalities at once. PlotThe Plot command, from the Graphs section, will plot any inequality involving two variables. In order to plot the region satisfied by a single inequality involving x and y, go to the basic inequality plotting page, where you can enter the inequality and specify the upper and lower limits on x and y that you want the graph to be plotted for. The advanced inequality plotting page allows you to plot the union or intersection of up to 8 regions on the one graph. You have control over such things as whether or not to show the axes, where the axes should be located and what the aspect ratio of the plot should be. In addition, you have the option of showing each individual region on its own. Introduction to InequalitiesAn 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, another. As with equations, a value of the variable for
which the inequality is true is a solution of the inequality, and the set of all
such solutions is the solution set of the inequality. Two inequalities with the
same solution set are equivalent inequalities. Inequalities are solved with the
following properties of inequality. For real numbers a, b, and c: (The same number may be added to both sides of an inequality without changing the solution set.)
(Both sides of an inequality may be multiplied by the same positive number without changing the solution set.)
Pay careful attention to part (c): if both sides of an inequality are multiplied by a negative number, the direction of the inequality symbol must be reversed. For example, starting with the true statement  3 < 5 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 true result only if the
direction of the inequality symbol is reversed.
Now multiply both sides by 1/3. (We could also divide by 3.) Since 1/3 < 0, reverse the direction of the inequality symbol. The original inequality is satisfied by any real number less than 4. The solution set can be written {xx < 4}. A graph of the solution set is shown in Figure 2.6, where the parenthesis is used to show that 4 itself does not belong to the solution set.
Example 2
Solve 2 < 5 + 3m < 20.
