Solving Linear Inequalities
When working with linear equations involving one variable whose highest degree (or order) is one, you are looking for THE one value of the variable that will make the equation true. But if you consider an inequality such as x + 2 < 7, then values of x can be 0, 1, 2, 3, any negative number, or any fraction in between. In other words, there are many solutions for this inequality. Fortunately, solving an inequality involves the same strategies as solving a one variable equation. So even though there are an infinite number of answers to an inequality, you do not have to work any harder to find the answer. To review how to solve one variable equations, click here. (linear equations solving.doc)

However, there is one major difference that you must keep in mind when working with any inequality. If you multiply or divide by a negative number, you must change the direction of the inequality sign. You’ll see why this is the case soon.

Let’s go back and look at x + 2 < 7. If this were an equation, you would only need to subtract 2 from both sides to have x by itself.
Keep in mind that the new rule for inequalities only applies to multiplying or dividing by a negative number. You can still add or subtract without having to worry about the sign of the inequality.

But what would happen if you had ? Before solving, let’s think about some values of x that will make this inequality true. If you let x = -5 or -6 or any other value that is less than -5, then the inequality will be true. So you would write your solution as . In the process of solving this inequality using algebraic methods, you would have something that looks like the following.

Let's Practice
Begin by getting the variable on one side by itself by subtracting 3 from both sides. Then divide both sides by 2. Since you are dividing by a positive 2, there is no need to worry about changing the sign of the inequality.
The solution to this problem begins with subtracting 4 from both sides and then dividing by -3. As soon as you divide by -3, you MUST change the sign of the inequality.
This solution will require a little more manipulation than the previous examples. You have to gather the terms with the variables on one side and the terms without the variables on the other side.
There is another type of inequality called a double inequality. This is when the variable appears in the middle of two inequality signs. This is simply a shortcut way of writing two separate inequalities into one and using a shorter process for finding the solution.

More Practice:
The strategy for solving this inequality is not that much different than the other examples. Except in this case, you are trying to isolate the variable in the middle rather than on one side or the other. But the process for getting the x by itself in the middle, you should add 1 to all three parts of the inequality and then divide by 6.
When each of these problems has been solved, the final answer has been given as an inequality. If your teacher or your textbook also asks you to display your answer graphically, click here to learn how this can be done.

Examples
Solve each inequality.