Algebra/Equalities and Inequalities

Solving linear inequalities involves finding solutions to expressions where the quantites are not equal.

A number on the number line is always greater than any number on its left and smaller than any number on its right. The symbol "<" is used to represent "is less than", and ">" to represent "is greater than".

For example:

<--|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|----->
  -5    -4    -3    -2    -1     0     1     2     3     4     5

From the number line, we can easily tell that 3 is greater than -2, because 3 is on the right side of -2 (or -2 is on the left of 3). We write it as $3>-2$ (or as $-2<3$). We can also derive that any positive number is always greater than negative number.

Consider any two numbers, a and b. One and only one of the following statements can be true:

1. $a>b$,
2. $a=b$, or
3. $a<b$

This is the Law of Trichotomy.

For an inequality with one unknown, there may be many (sometimes infinite) possible solutions.

Properties

• Transitive property:

For any three numbers $x$, $y$, $z$, if $x>y$ and $y>z$, then $x>z$.

• Additive property:

In an inequality, we can add or subtract the same value from both sides, without changing the sign (i.e. ">" or "<"). That is to say, for any three numbers $x$, $y$ and $p$, if $x>y$, then $x+p>y+p$ and $x-p>y-p$.

• Multiplicative property

We can multiply or divide both sides by a positive number without changing the sign. For example, if we have any two numbers $x$ and $y$, and another positive number $p$, then if $x>y$, then $x\cdot p>y\cdot p$ and $\frac{x}{p}>\frac{y}{p}$.

When we multiply or divide both sides by a negative number, we have to change the sign of the inequality (i.e, ">" change to "<" and vice versa). So if we are given two numbers $x$ and $y$, and another negative number $p$, then if $x>y$, $x\cdot p<y\cdot p$ and $\frac{x}{p}<\frac{y}{p}$.

Now we can go on to solve any linear inequalities.

Solving Inequalities

Solving inequalities is almost the same as solving linear equations. Let's consider an example: $x+4<13$. All we have to do is to subtract 4 on both sides. We will then get $x<9$, and that is the answer! Note, however, what you get is not a single answer, but a set of solutions, i.e., any number that satisifies the condition $x<9$ (any number that is less than 9) can be a solution to the inequality. It is very convenient to represent the solution using the number line:

<-------------------o
<-+-----+-----+-----+-----+-----+-->
  6     7     8     9     10    11

(Note: the open circle ("o") shows that the value 9 is not included in the solution set, as the inequality of this equation is less than 9, not less than or equal to 9. When we deal with less (greater) than or equal to (≤ or ≥) later on, we use a closed circle ("●") to show that the value is included in the solution set.)

Let us try another more complicated question: $3x-2\ge 2(x-3)$. First, you may want to expand the right hand side: $3x-2\ge 2x-6$. Then we can simply rearrange the terms so that all the unknown variables are on one side of the equation, usually the left hand side: $3x-2x\ge -6+2$. Hence we can easily get the answer: $x\ge -4$. This solution is represented on the number line below. Note that the solution requires a closed circle ("●"), because the $x$ is greater than or equal to 4.

              ●------------------->
<-+-----+-----+-----+-----+-----+-->
 -6    -5    -4    -3    -2    -1

Compound Inequalities

A compound inequality is a pair of inequlities related by the words and or or. In an and inequality, both inequalities must be satisfied. All possible solution values will be located between two defined numbers, and if this is impossible, the compound inequality simply has no solutions.

Consider this example: $x+6\ge 2$ and $x\le 2$. First, solve the first inequality for x to get $x\ge -4$. All and inequalities can be rewritten as one inequality, like this: $-4\le$$x\le 2$ (write x between two ≤'s or <'s or both with the smaller number on the left and the larger number on the right). Now, we can graph this inequality on a number line as a line segment. Remember, all solutions to ≤ or ≥ must be graphed with closed circles. Interpret this graphic as "all numbers between -4 and 2, including -4 and 2."

        ●-----------------●
<-+-----+-----+-----+-----+-----+-->
 -6    -4    -2     0     2     4

Now, let us consider or inequalities. Or inequalities usually do not have a set of solutions that satisfies both. Instead, they usually have two sets of infinite numbers that are solutions to each one. Because of this, or graphs define which numbers satisfy either equation. For example: $x<1$ or $x-1\ge 2$. First, solve for x in the second inequality to get $x\ge 3$. Now, graph the two inequalities on the same number line. Remember to use open and closed circles accordingly.

<-------------o           ●-------->
<-+-----+-----+-----+-----+-----+-->
 -1     0     1     2     3     4

Special Cases

Sometimes or inequalities can be simplified into one non-compound inequality. Consider this example: $x\ge 2$ or $-3x\le -12$. Solve the second inequality for x by dividing each side by -3. Since the inequality sign is reversed when both sides are multiplied or divided by a negative, the inequality solved for x is $x\ge 4$. When the inequalities are graphed, they overlap and point in the same direction:

              ●---------->●-------->
<-+-----+-----+-----+-----+-----+-->
  0     1     2     3     4     5

Thus, the compound inequality is equivalent to simply $x\ge 2$. The true graph of this compound inequality should look like this, with the closed circle for $x\ge 4$ not shown since that specific solution is included in the line that follows 2:

              ●-------------------->
<-+-----+-----+-----+-----+-----+-->
  0     1     2     3     4     5

---

There is also another special case that pertains to certain or inequalities. Consider the example $2x\le -2$ or $2x+1\ge -5$. Solve each equation for x to get $x\le -1$ or $x\ge -3$. The graph for these two equations would look like this:

<-------------●-----------●-------->
<-+-----+-----+-----+-----+-----+-->
 -5    -4    -3    -2    -1     0

Since the graph of the solutions encompasses all the numbers on the number line, the solution to this compound inequality is simply "all real numbers." That is, every number to infinity is a possible solution to this compound inequality.

Solving Inequalities with Absolute Value

Since $|x|=|-x{|}^{‴}$ A inequality involving absolute value will have to solved in two parts.

Solving $|x-6|<5^{‴}$

The first part would be $x-6<5^{‴}$ which gives $x<11$. The second part would be $-(x-6)<5^{‴}$ which solved yeilds $x>1$.

So the answer to $|x-6|<5$ is $1<x<11$

        ●----------------------------><-----------------------------●
<-+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-->
  0     1     2     3     4     5     6     7     8     9     10    11    12

Graphing Linear Inequalties

The graphing of linear inequalities is very similar to the graphing of linear functions. A linear inequality is written in

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