Intermediate Algebra/Solving Inequalities

Inequalities.
As opposed to an equation, an inequality is an expression that states that two quantities are unequal or not equivalent to one another. In most cases we use inequalities in real life more than equations (i.e. this shirt costs $2 "more than" that one).
Given that a and b are real numbers, there are four basic inequalities:
Properties of Inequalities.
Just as there are four properties of equality, there are also four properties of inequality:
Addition Property of Inequality
If a, b, and c are real numbers such that a &gt; b, then a + c &gt; b + c. Conversely, if a &lt; b, then 
a + c &lt; b + c.
Subtraction Property of Inequality
If a, b, and c are real numbers such that a &gt; b, then a - c &gt; b - c. Conversely, if a &lt; b, then
a - c &lt; b - c.
Multiplication Property of Inequality
If a, b, and c are real numbers such that a &gt; b and c &gt; 0, then ac (or a * c) &gt; bc (or b * c). Conversely, if a &lt; b and c &gt; 0, then ac &lt; bc. (Note that if c = 0, then both sides of the inequality are in fact equal.) We will also review cases where c is less than zero later in the lesson.
Division Property of Inequality
If a, b, and c are real numbers such that a &gt; b, and c &gt; 0, then formula_5. Conversely under the same conditions, if a &lt; b, then formula_6. As with the Multiplication Property, there are special cases that will be discussed later when c &lt; 0.
Note that all four properties also work with inequalities where formula_1 or formula_3.
Trichotomy Property.
The statements above form the basis of Trichotomy Property:
 Given any two real numbers a and b, then only one of the following statements must hold true:
So, if we are given any two unknown real-number values, then any one of the three statements will hold true.
Solving Inequalities.
Solving algebraic inequalities is more or less identical to solving algebraic equations. Consider the following example:
 formula_9
Although it may be an inequality, we can use the Properties of Inequality stated above to solve. Start by subtracting 2x from both sides:
 formula_10
 formula_11
Finish by adding 7 to both sides.
 formula_12 
 formula_13
This can be rewritten as formula_14. To check, substitute any value greater than or equal to 11. However, in order to satisfy the Trichotomy Property, we'll substitute three different values: 10, 11, and 12.
Ten is incorrect, whereas eleven and twelve satisfy the solution. Therefore, the solution set - the set of all answers which satisfy the original inequality - is formula_14. Written in set notation, the answer is {formula_16}. This is read as "the set of all x such that x is greater than or equal to 11".
Special Cases - A variable in the denominator.
For example, consider the inequality
In this case one cannot multiply the right hand side by (x-1) because the value of x is unknown. Since x may be either positive or negative, you can't know whether to leave the inequality sign as &lt;, or reverse it to &gt;.
The method for solving this kind of inequality involves four steps:
Practice Problems.
&lt;quiz display=simple points="1/1"&gt;
+yes
-no
-yes
+no
-yes
+no
&lt;/quiz&gt;
Special Cases.
Let us suppose that we were given this inequality to solve:
 formula_29
Using the same steps as above, start by subtracting 8 from both sides.
 formula_30
 formula_31
Now divide both sides by -3.
 formula_32
 formula_33
Check this solution by substituting three numbers. We'll use -7, -6, and -5.
Wait, what happened? -7 and -6 satisfy the inequality, yet -7 is non-inclusive in the solution set! And -5, which is greater than -6, does not satisfy the inequality!
This is a special case involved in solving inequalities. Because the coefficient of the x-term was negative, the constant on the other side (26, which became 18 and then -6) switched signs. In order to attain a valid solution if a negative number is divided, we need to "switch the sign" in order to make sure that the solution set is correct. Thus, formula_33 becomes formula_35, or more specifically, {formula_36}.
&lt;hr&gt;
Practice Problems.
&lt;quiz display=simple points="1/1"&gt;
&lt;/quiz&gt;
Graphing Solutions.
Because inequalities have multiple solutions, we need to be able to represent them graphically. In order to do so, we use the number line, depicted below:
 -5 -4 -3 -2 -1 0 1 2 3 4 5
There are two ways of graphing solutions; however, each one is unique depending on the nature of the inequality. If, for example, the inequality contains the &lt; or &gt; sign, we use an open circle ("O") and place it on (or above) the corresponding position on the number line, then draw another line either left or right of the solution (based on the sign) to indicate the infinite number of solutions in the set. For example, if we were to graph x &lt; 4:
 &lt;-----------------------------O
 -5 -4 -3 -2 -1 0 1 2 3 4 5
The open circle indicates that 4 is not included in the solution set; however, all values less than 4 satisfy the solution.
If an inequality contains a formula_37 or formula_40, then a closed circle (it will be depicted here with a *) is placed on or above the corresponding position on the number line. Hence, the solution formula_41 is graphed as follows.
 &lt;--|--|--|--|--|--|--|--|--|--|--|--&gt;
 -5 -4 -3 -2 -1 0 1 2 3 4 5
The asterisk (*) indicates that -2 is inclusive in the solution set, as are all values greater than -2.
Practice Problems
Graph the following inequalities on a number line:
1. x &gt; 4
2. formula_42
3. -1 &gt; x
Solutions
1. 
 O-----&gt;
 -5 -4 -3 -2 -1 0 1 2 3 4 5
2. 
 &lt;--|--|--|--|--|--|--|--|--|--|--|--&gt;
 -5 -4 -3 -2 -1 0 1 2 3 4 5
3. Be careful here; you need to rearrange the inequality first before graphing. When rewritten, the inequality becomes x &lt; -1:
 &lt;--------------O
 -5 -4 -3 -2 -1 0 1 2 3 4 5
Lesson Review.
An inequality is a statement justifying that two quantities are not equal to each other. There are four cases of inequalities, two of which allow for equality (formula_1 and formula_3). The four properties of inequality, which are more or less parallel to the properties of equality, can be used to solve simple inequalities. The only exceptions are in multiplication and division, where the signs must be reversed if both sides are multiplied or divided by a negative number. Finally, the solution set of an inequality can be graphed on the number line with either an open circle ("O") or a closed circle ("*"), depending on the original sign used.
Lesson Quiz.
1. What is an inequality? List the four possible cases of inequalities.
2. State the Trichotomy Property in your own words.
3. Solve the following and graph all solutions:
 a. 4x + 3 &gt; 7 b. formula_45
 c. formula_46 
 d. -x + 4 &gt; 3x
Quiz Answers.
1. An inequality states that two quantities are not equal. The four cases of inequalities: a &lt; b, a &gt; b, formula_1, formula_3.
2. Answers will vary. Just make sure you didn't copy and paste the actual definition.
3a. x &gt; 1
 O--------------&gt;
 -5 -4 -3 -2 -1 0 1 2 3 4 5
3b. formula_49
 &lt;--|--|--|--|--|--|--|--|--|--|--|--&gt;
 -5 -4 -3 -2 -1 0 1 2 3 4 5
3c. formula_50
 &lt;--|--|--|--|--|--|--|--|--|--|--|--|--|--&gt; 
 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
3d. x &lt; 1
 &lt;--------------------O
 -5 -4 -3 -2 -1 0 1 2 3 4 5