Solving Algebraic Inequalities

(1) Inequality WORKSHEETS

(2) Solving One Step Inequalities - Video Examples

(3) Solving Two Step Inequalities - Video Examples

(4) Solving Compound Inequalities - Video Examples

(5) Solving Absolute Value Inequalities - Video Examples

(6) Other Youtube Videos on Solving Inequalities

When solving inequalities it is important to realize most of the steps are identical to solving equations. The principles are the same:

1) Two Sides - It is important to mentally break the inequality into two parts, the left side and the right side. The inequality sign (< or >) is what seperates the left from the right side.

2) Variable by Itself - Remember the goal is always to solve the inequality such that the variable is by itself on one side of the inequality. "By itself" means that there is nothing being added or subtracted to the variable and that the coefficient is one. (Remember the coeffecient of a variable is the number in front of the variable or the number being multiplied by the variable)

3) Types of Inequalites - There are two types of one step inequalities:

a) Addition/Subtraction - these are problems where a number is being added or subtracted to the variable (x + 4 > 7 or 5 < y - 3 ). In these problems you "do the opposite" which means that if a number is being added to the variable then you subtract that number from both sides of the inequality.

b) Multiplication/Division - these problems are when a number(other than 1) is being multiplied or divided by number. To solve these you also "do the opposite", which means that if a variable is being multiplied by a number then you divide both sides by the number. If a variable is being divided by that number then you multiply by that number. VERY IMPORTANT - If the number you are dividing or multiplying by is negative then you must change the direction of the inequality sign(see example 4).

4) What is a Solution to Inequality? - When you have solved an inequality the solution is actually a set of numbers. x < 5 means all the numbers less than five. The way we designate this is by shading in all the numbers on a number line that are actually less that 5. If the inequality being used is < this is called less than or equal to. At the cutoff point(5 from the previous example) we place a closed circle, because part of the solution is five. 5<5 is a true statement because the are equal to each other. The following are examples of solving inequalities.

As with one step inequalities, two step inequalities are solved very similar to two step equations. Besides graphing the solution the only real difference is you when you divide by a negative you must change the direction of the inequality sign. My sister inlaw has her students say "woot woot" whenever the divide by a negative to help the remember that is when the change the direction. The steps are as follows:

1) Simplify both sides of the inequalities - Remember that you must think of the inequality as having two sides. Thus you first attempt to simplify the left(use adding like terms and distributive property) and then you simplify the right side.

2) Add or Subtract To Get the Variable Alone - Next you look at the side the variable is on. If it is a two step inequality then the variable is being multiplied by the number and something is being added/subtracted from the variable and the coefficient. The first step is to get rid of the number being added/subtracted from the variable. This is accomplished by doing the same thing you did in example 1 and 2 above.

3) Multiply or Divide to Finish - After completing step 2 above you must now take care of the coefficient of the variable. If the variable is being multiplied by the number then you divide and vice versa. At this point the problem looks exactly like examples 3 and 4 above and should be solved accordingly.

The following videos show examples of how to solve and graph inequalities.

In elementary school you learned that a compound word is two words that are put together to make one. A compound inequality is similar. It is a problem that uses two inequalities to find one solution. Remember that the solution to an inequality are all the numbers that make it true. The same is true of compound inequalities.

Two Types

Type 1 - "Or" Inequalities

These are compound inequalities connected by the word "or". x > 3 or x < -2. The answer to this problem is all the number that satisfy one of the two inequalities. Any number that is greater than 3 or less than -2.

These are solved simply by solving the two inequalities seperately. Once both are solved for the variable, graph them seperately on the SAME number line. "Or" inequalities usually have shading going in opposite directions.

Type 2 - "And" Inequalities

These are compound inequalities connected by the word "and". y > -2 and y < 4. The nature of the word "and" is different than "or". "And" implies that a number must satisfy BOTH inequalities to be a part of the solution. 3 > -2 and 3 < 4. Thus three is part of the solution set. "And" inequalities on a number line are usually represented by all the numbers in between the two cut-off(-2 and 4 from the above example) numbers.

There is also another way that "and" inequalities can be written. 4 < x < 8. One way to deal with this is to rewrite it as two seperate inequalities. 4 < x and x < 8. The other way is to interpret it as x is between 4 and 8.

"And" inequalities can be solved just like or inequalities. Thus you solve the two inequalities seperately. However, if they are put together like this 3 < 2x + 1 < 7, then you consider the inequality to have three sides rather than two. A left, middle and right side. To solve this you look at the middle and whatever you do to the middle you must do to the left and right side. For instance on this problem you subtract 1 from the middle to get 2x by itself. Thus we also must subtract 1 from 3 and 7 so it would be rewritten 2 < 2x < 6.

Try the following examples and then watch the videos to see the solutions.

The last type of single variable inequalities involve absolute values. Absolute value inequalities can be considered compound inequalities. There are some very specific rules about how to REWRITE an absolute value inequalitiy as a compound inequality.

Rule 1

a) The absolute value of something greater than a number (|x| > a) is always an "or" inequality.

b) The absolute value of something less than a number (|x| < a) is always an "and" inequality.

Rule 2

Rewrite the absolute value inequalities as two seperate inequalities. The first inequality looks the same as the original without the absolute value symbol. Then to find the second you change the direction of the inequality sign and change the sign of the second number.

So |x + 3| > 2 is rewritten as:

x + 3 > 2 or x + 3 < -2

And |x + 4| < 5 is rewritten as:

x + 4 < 5 and x + 4 > -5

You finish the problem by treating them as compound inequalities that are described above.