Mathematics: Inequalities

Working with Inequalities
Commonly in algebra and other upper-level fields of math you find yourself working with mathematical expressions that do not equal each other. The statement comparing such expressions with symbols such as < (less than) or > (greater than) is called an inequality.

An example of an inequality is
.
To solve for
, simply divide both sides by

and the solution is shown to be
.

Graphs of the solution set of inequalities are represented on a number line. Open circles are used to show that an expression approaches a number but is never quite equal to that number.

Conditional inequalities are those with certain values for the variable that will make the condition true and other values for the variable where the condition will be false. Absolute inequalities can have any real number as the value for the variable to make the condition true, while there is no real number value for the variable that will make the condition false. Solving inequalities is done by following the same rules for solving equations with the exception that when multiplying or dividing by a negative number the direction of the inequality sign must be flipped or reversed. Double inequalities are situations where two inequality statements apply to the same variable expression. An example of this is
.

Determining Solutions to Inequalities
To determine whether a coordinate is a solution of an inequality, you can substitute the values of the coordinate into the inequality, simplify, and check whether the resulting statement holds true. For instance, to determine whether

is a solution of the inequality
, substitute the values into the inequality,
. Simplify the right side of the inequality and the result is
, which is a false statement. Therefore, the coordinate is not a solution of the inequality. You can also use this method to determine which part of the graph of an inequality is shaded. The graph of

includes the solid line
and, since it excludes the point

to the left of the line, it is shaded to the right of the line.

Flipping Inequality Signs
When given an inequality, we can always turn the entire inequality around, swapping the two sides of the inequality and changing the inequality sign. For instance,
applies to multiplication and division, and only with negative numbers.<br> Multiplying or dividing both sides by a positive number, or adding or subtracting any number regardless of sign, does not flip the inequality.<br> <br> Compound Inequalities<br> A compound inequality is an equality that consists of two inequalities combined with <i>and</i> or <i>or</i>. The two components of a proper compound inequality must be of opposite type: that is, one must be greater than (or greater than or equal to), the other less than (or less than or equal to). For instance, " />
or
' is a compound inequality, as is '

and
.' A. and inequality can be written more compactly by having one inequality on each side of the common part: '

and
,' can also be written as
.
In order for the compound inequality to be meaningful, the two parts of an and inequality must overlap; otherwise no numbers satisfy the inequality. On the other hand, if the two parts of an or inequality overlap, then all numbers satisfy the inequality and as such is usually not meaningful.
Solving a compound inequality requires solving each part separately. For example, given the compound inequality '

or
,' the first inequality,
, reduces to
, and the second part,
, reduces to
, so the whole compound inequality can be written as '

or
.' Similarly,

can be solved by dividing each term by 2, yielding
.

Solving Inequalities Involving Absolute Values
To solve an inequality involving an absolute value, first isolate the term with the absolute value. Then proceed to treat the two cases separately as with an absolute value equation, but flipping the inequality in the case where the expression in the absolute value is negative (since that essentially involves multiplying both sides by
.)
The two cases are then combined into a compound inequality; if the absolute value is on the greater side of the inequality, then it is an or compound inequality, if on the lesser side, then it's an and.

Consider the inequality
. We can isolate the absolute value term by subtracting 2 from both sides:
. Now, we're left with the two cases

or
: note that in the latter, negative case, the inequality is flipped.

reduces to
, and

reduces to
. Since in the inequality

the absolute value is on the greater side, the two cases combine into an or compound inequality, so the final, solved inequality is '

or
.'

Solving Inequalities Involving Square Roots
Solving an inequality with a square root involves two parts.
First, we solve the inequality as if it were an equation, isolating the square root and then squaring both sides of the equation. Second, we restrict the solution to the set of values of

for which the value inside the square root sign is non-negative.
For example, in the inequality,
, we can isolate the square root by subtracting 1 from both sides, yielding
. Squaring both sides of the inequality yields
, so
.
Since we can't take the square root of a negative number, we also require the part inside the square root to be non-negative. In this case, that means
.
Adding 2 to both sides of the inequality yields
.
Our final answer is a compound inequality combining the two simple inequalities:

and
, or
.
Note that we only get a compound inequality if the two simple inequalities are in opposite directions; otherwise we take the one that is more restrictive.
The same technique can be used for other even roots, such as fourth roots. It is not, however, used for cube roots or other odd roots—negative numbers do have cube roots, so the condition that the quantity inside the root sign cannot be negative does not apply.

Special Circumstances
Sometimes an inequality involving an absolute value or an even exponent is true for all values of , and we don't need to do any further work to solve it. This is true if the inequality, once the absolute value or exponent term is isolated, says that term is greater than a negative number (or greater than or equal to zero). Since an absolute value or a number raised to an even exponent is always non-negative, this inequality is always true.

Graphical Solutions to Equations and Inequalities
When equations are shown graphically, they are usually shown on a Cartesian coordinate plane. The Cartesian coordinate plane consists of two number lines placed perpendicular to each other and intersecting at the zero point, also known as the origin. The horizontal number line is known as the -axis, with positive values to the right of the origin, and negative values to the left of the origin. The vertical number line is known as the y-axis, with positive values above the origin, and negative values below the origin. Any point on the plane can be identified by an ordered pair in the form , called coordinates. The x-value of the coordinate is called the abscissa, and the y-value of the coordinate is called the ordinate. The two number lines divide the plane into four quadrants: I, II, III, and IV.

Note that in quadrant I

and
, in quadrant II

and
, in quadrant III

and
, and in quadrant IV

and
.
Recall that if the value of the slope of a line is positive, the line slopes upward from left to right. If the value of the slope is negative, the line slopes downward from left to right. If the y-coordinates are the same for two points on a line, the slope is 0 and the line is a horizontal line. If the x-coordinates are the same for two points on a line, there is no slope and the line is a vertical line. Two or more lines that have equivalent slopes are parallel lines.
Perpendicular lines have slopes that are negative reciprocals of each other, such as  and .

Graphing Simple Inequalities
To graph a simple inequality, we first mark on the number line the value that signifies the end point of the inequality. If the inequality is strict (involves a less than or greater than), we use a hollow circle; if it is not strict (less than or equal to or greater than or equal to), we use a solid circle. We then fill in the part of the number line that satisfies the inequality: to the left of the marked point for less than (or less than or equal to), to the right for greater than (or greater than or equal to).
For example, we would graph the inequality  by putting a hollow circle at 5 and filling in the part of the line to the left:


Graphing Compound Inequalities
To graph a compound inequality, we fill in both parts of the inequality for an or inequality, or the overlap between them for an and inequality. More specifically, we start by plotting the endpoints of each inequality on the number line. For an or inequality, we then fill in the appropriate side of the line for each inequality. Typically, the two component inequalities do not overlap, which means the shaded part is outside the two points. For an and inequality, we instead fill in the part of the line that meets both inequalities.

For the inequality '

or
,' we first put a solid circle at –3 and a hollow circle at 4. We then fill the parts of the line outside these circles:


Graphing Inequalities Including Absolute Values

An inequality with an absolute value can be converted to a compound inequality. To graph the inequality, first convert it to a compound inequality, and then graph that normally. If the absolute value is on the greater side of the inequality, we end up with an or inequality; we plot the endpoints of the inequality on the number line and fill in the part of the line outside those points. If the absolute value is on the smaller side of the inequality, we end up with an and inequality; we plot the endpoints of the inequality on the number line and fill in the part of the line
between those points.
For example, the inequality

can be rewritten as

or

We place solid circles at the points 3 and

and fill in the part of the line outside them:


Graphing Equations in Two Variables
One way of graphing an equation in two variables is to plot enough points to get an idea for its shape, and then draw the appropriate curve through those points. A point can be plotted by substituting in a value for one variable and solving for the other. If the equation is linear, we only need two points, and can then draw a straight line between them. equation
. This is a linear equation—both variables only appear raised to the first power—so we only need two points. When
,
. When
,
. We can therefore choose the points

and
, and draw a line between them:


Graphing Inequalities in Two Variables
To graph an inequality in two variables, we first graph the border of the inequality. This means graphing the equation that we get if we replace the inequality sign with an equals sign. If the inequality is strict
(> or <), we graph the border with a dashed or dotted line; if it is not strict (≥ or ≤), we use a solid line. We can then test any point not on the border to see if it satisfies the inequality. If it does, we shade in that side of the border; if not, we shade in the other side. As an example, consider
true, so we shade in the side of the border that does <i>not</i> include the point <br><img data-cke-saved-src=" />:


Graphing Compound Inequalities in Two Variables
One way to graph a compound inequality in two variables is to first graph each of the component inequalities. For an and inequality, we then shade in only the parts where the two graphs overlap; for an or inequality, we shade in any region that pertains to either of the individual inequalities.

Consider the graph of '
:
We first shade in the individual inequalities:


Now, since the compound inequality has an and, we only leave shaded the overlap—the part that pertains to both inequalities:

If instead the inequality had been '
,' our final graph would involve the total shaded area:


P1. Analyze the following inequalities:
(a)


(b)

P2. Graph the following on a number line:
(a)


(b)


(c)

P1. (a) Subtracting 2 from both sides yields
; multiplying by
—and flipping the inequality, since we're multiplying by a negative number—yields
. But since the absolute value cannot be negative, it's always greater than –1, so this inequality is true for all values of
.
(b) Subtracting 7 from both sides yields
; dividing by
2 yields
. But

must be nonnegative, and hence cannot be less than or equal to –3; this inequality has no solution.
P2. (a) We would graph the inequality

by putting a solid circle at 3 and filling in the part of the line to the right:

(b) The inequality

is equivalent to '

and
.'
To plot this compound inequality, we first put solid circles at –2 and 6, and then fill in the part of the line between these circles:

(c)
The inequality

can be rewritten as '

and
.'
We place hollow circles at the points –2 and 2 and fill in the part of the line between them: