A linear inequality is the same as a linear equation, except it has an inequality symbol instead of an = symbol. This means that we want to find all numbers that make one side of the expression greater than or less than the other side. As a result we may have more than one number that will make the inequality true. Therefore we will graph our "solution set" on a number line. To solve a linear inequality, we will follow the same steps as when we solve a linear equation, except whenever we multiply or divide the sides by a negative number, we change the direction of the inequality. This is due to the properties of negative numbers: 4 > 2 but -4 < -2.
For example, if we have: -3x < 12, when we divide both sides by -3, we get x > - 4.

Now let us try an example.

We want to get x by itself. We will start by subtracting 2x from both sides, then add 3 to both sides.

Now we can graph this on a number line. Since our inequality is less than or equal to 4, we will start at four with a closed circle (illustrating the equality) and shade toward the smaller numbers (left). The numbers that are shaded are the numbers that will make the inequality true (the solution set).
 

Compound Inequalities

Sometimes we have to consider the solutions sets of two inequalities at the same time. Such inequlities are called compound inequalities and consist of two inequalities connected by logical operations "AND" or "OR". The logical operation "and" tells us to look at the intersection (or the overlap) of the two solution sets. The logical operation "or" tells us to look at the union (or the combination) of the two solution sets. For example, if we have the set A = {a, b, c, d, e} and the set B = {d, e, f, g}, then A "AND" B is the set {d, e} since those are the elements they have in common. Using the same sets, A "OR" B is the set {a, b, c, d, e, f, g}, which is the combination of the two sets.

 Therefore if we have inequalities x £ 4 "AND" x > -1, the intersection of the two solution sets is going to be the numbers between -1 and 4 (-1 < x £ 4).

Notice that it is possible for the two sets not to overlap at all. In such cases, the solution set is the empty set.
 
 

If we take the same inequalities as above and connect them with an "OR" (x £ 4 "OR" x >-1), then we are combining the two solution sets which results in all real numbers. That is, the whole number is shaded.
 

It helps to graph each solution set separately. In the case of an "AND" we look for the overlapping part of the two graphs. In the case of an "OR" the solution consists of any number that is shaded by either graph.

Now you try a few

A.

Solution
 
 

B. (Hint: get x by itself in the middle!!)

Solution
 
 

C.

 x < 5 AND x < 2
 
 

Solution
 
 

D.

 x < 5 OR x < 2
 
 

Solution
 
 

Solutions:

A.

Now we will graph our answer.
 

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B.

Now we graph our solution.

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C.
 
 

Note: the solution set is only the red part. The other two parts are just there to help us visualize.

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D.
 
 

Note: the solution set is only the red part. The other two parts are just there to help us visualize.

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