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Beginning Algebra
Tutorial 18: Solving Linear Inequalities

Learning Objectives

 After completing this tutorial, you should be able to: Use the addition, subtraction, multiplication, and division properties of  inequalities to solve linear inequalities. Draw a graph to give a visual answer to an inequality problem.

Introduction

 When solving linear inequalities, we use a lot of the same concepts that we use when solving linear equations.  Basically, we still want to get the variable on one side and everything else on the other side by using inverse operations.  The difference is, when a variable is set equal to one number, that number is the only solution.  But, when a variable is less than or greater than a number, there are an infinite number of values that would be a part of the answer.

Tutorial

 Inequality Signs

 Read left to right: a < b    a is less than b a < b   a is less than or equal to b a > b     a is greater than b a > b    a is greater than or equal to b

 Graphing Inequalities

x < c

 When x is less than a constant, you darken in the part of the number line that is to the left of the constant.  Also, because there is no equal line, we are not including where x is equal to the constant.  That means we are not including the endpoint.  One way to notate that is to use an open hole at that point.

x > c

 When x is greater than a constant, you darken in the part of the number line that is to the right of the constant.  Also, because there is no equal line, we are not including where x is equal to the constant.  That means we are not including the endpoint.  One way to notate that is to use an open hole at that point.

x < c

 When x is less than or equal to a constant, you darken in the part of the number line that is to the left of the constant.  Also, because there is an equal line, we are including where x is equal to the constant.  That means we are  including the endpoint.  One way to notate that is to use an closed hole at that point.

x > c

 When x is greater than or equal to a constant, you darken in the part of the number line that is to the right of the constant.  Also, because there is an equal line, we are including where x is equal to the constant.  That means we are including the endpoint.  One way to notate that is to use a closed hole at that point.

 Example 1:  Graph x > 5.

 Since we needed to indicate all values greater than 5,  the part of the number line that is to the right of 5 was darkened. Since there is no equal line under the > symbol, this means we do not include the endpoint 5 itself.  We can notate that by using an open hole (or you can use a curved end).

 Example 2:  Graph x < 2.

 Since we needed to indicate all values less than or equal to 2,  the part of the number line that is to the left of 2 was darkened. Since there is an equal line under the < symbol, this means we do include the endpoint 2.  We can notate that by using a closed hole (or you can use a boxed end).

 Addition/Subtraction Property for Inequalities If a < b, then a + c < b + c If a < b, then a - c < b - c

 In other words, adding or subtracting the same expression to both sides of an inequality does not change the inequality.     Example 3:  Solve the inequality and graph the solution set.

 Graph: *Inv. of sub. 7 is add. 7       *Visual showing all numbers less than 4 on the number line

 Note that the inequality stayed the same throughout the problem.  Adding or subtracting the same value to both sides does not change the inequality. The answer 'x is less than 4' means that if we put any number less than 4 back in the original problem, it would be a solution (the left side would be less than the right side).  As mentioned above, this means that we have more than just one number for our solution, there are an infinite number of values that would satisfy this inequality. Graph:  Since we needed to indicate all values less than 4, the part of the number line that was to the left of 4 was darkened.  Since we are not including where it is equal to, an open hole was used.

 Example 4: Solve the inequality and graph the solution set.

 Graph: *Inv. of add 10 is sub. 10       *Visual showing all numbers greater than or = to -5 on the number line.

 Note that the inequality stayed the same throughout the problem.  Adding or subtracting the same value to both sides does not change the inequality. The answer 'x is greater than or equal to -5' means that if we put any number greater than or equal to -5 back in the original problem, it would be a solution (the left side would be greater than or equal to the right side).  As mentioned above, this means that we have more than just one number for our solution, there are an infinite number of values that would satisfy this inequality. Graph:  Since we needed to indicate all values greater than or equal to -5, the part of the number line that was to the right of -5 was darkened. Since we are including where it is equal to, a closed hole was used.

 Multiplication/Division Properties for Inequalities when multiplying/dividing by a positive value If a < b  AND  c is positive, then   ac < bc If a < b  AND  c is positive, then   a/c < b/c

 In other words, multiplying or dividing the same POSITIVE number to both sides of an inequality does not change the inequality.     Example 5:  Solve the inequality and graph the solution set.

 Graph: *Inv. of mult. by 5 is div. by 5         *Visual showing all numbers less than -2 on the number line

 Note that the inequality sign stayed the same direction.  Even though the right side was a -10, the number we were dividing both sides by, was a positive 5.  Multiplying or dividing both sides by the same positive value does not change the inequality. Graph:  Since we needed to indicate all values less than -2, the part of the number line that was to the left of -2 was darkened. Since we are not including where it is equal to, an open hole was used.

 Example 6:  Solve the inequality and graph the solution set.

 Graph: *Inv. of div. by 3 is mult. by 3         *Visual showing all numbers greater than 3 on the number line

 Multiplying or dividing both sides by the same positive value does not change the inequality. Graph:  Since we needed to indicate all values greater than 3, the part of the number line that was to the right of 3 was darkened. Since we are not including where it is equal to, an open hole was used.

 Multiplication/Division Properties for Inequalities when multiplying/dividing by a negative value If a < b  AND  c is negative, then   ac > bc If a < b  AND  c is negative, then   a/c > b/c

 In other words, multiplying or dividing the same NEGATIVE number to both sides of an inequality reverses the sign of the inequality. The reason for this is, when you multiply or divide an expression by a negative number, it changes the sign of that expression.  On the number line, the positive values go in a reverse or opposite direction than the negative numbers go, so when we take the opposite of an expression,  we need to reverse our inequality to indicate this.   Example 7:  Solve the inequality and graph the solution set.

 Graph: *Inv. of div. by -2 is mult. by -2, so reverse inequality sign         *Visual showing all numbers less than -14 on the number line

 I multiplied by a -2 to take care of both the negative and the division by 2 in one step.  In line 2, note that when I did show the step of multiplying both sides by a -2, I reversed my inequality sign. Graph:  Since we needed to indicate all values less than -14, the part of the number line that was to the left of -14 was darkened. Since we are not including where it is equal to, an open hole was used.

 Example 8:  Solve the inequality and graph the solution set.

 Graph: *Inv. of mult. by -3 is div. by -3, so reverse inequality sign     *Visual showing all numbers greater than or = -3 on the number line

 In line 2, note that when I did show the step of dividing both sides by a -3, that I reversed my inequality sign. Graph:  Since we needed to indicate all values greater than or equal to -3, the part of the number line that was to the right of -3 was darkened. Since we are including where it is equal to, a closed hole was used.

 Strategy for Solving a Linear Inequality

Step 1:  Simplify each side, if needed.

 This would involve things like removing ( ), removing fractions, adding like terms, etc.

Step 2:  Use Add./Sub. Properties to move the variable term on one side and all other terms to the other side.

Step 3: Use Mult./Div. Properties to remove any values that are in front of the variable.

Note that it is the same basic concept we used when solving linear equations as shown in Tutorial 14: Solving Linear Equations.

Example 9: Solve the inequality and graph the solution set.

 Graph: *Inv. of sub. 3 is add. 3   *Inv. of mult. by -3 is div. both sides by -3, so reverse inequality sign       *Visual showing all numbers greater than -3 on the number line

 Graph:  Since we needed to indicate all values greater than -3, the part of the number line that was to the right of -3 was darkened. Since we are not including where it is equal to, an open hole was used.

 Example 10:  Solve the inequality and graph the solution set.

 Graph: *Distributive property *Get x terms on one side, constants on the other side *Inv. of mult. by 2 is div. by 2             *Visual showing all numbers less than -1/2 on the number line.

 Even though we had a -2 on the right side in line 5, we were dividing both sides by a positive 2, so we did not change the inequality sign.  Graph:  Since we needed to indicate all values less than -1/2, the part of the number line that was to the left of -1/2 was darkened. Since we are not including where it is equal to, an open hole was used.

 Example 11:  Solve the inequality and graph the solution set.

 Graph: *Mult. both sides by LCD   *Get x terms on one side, constants on the other side *Inv. of mult. by -1 is div. by -1, so reverse inequality sign         *Visual showing all numbers less than or equal to 4 on the number line.

 Once again we find ourselves dividing both sides by a negative value, as shown in line 6.  Once we do that, we need to remember to change the inequality.  Note that we still keep the equal part of it.  Graph:  Since we needed to indicate all values less than or equal to 4, the part of the number line that was to the left of 4 was darkened. Since we are including where it is equal to, a closed hole was used.

Practice Problems

 These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problems 1a - 1c: Solve the inequality and graph the solution set.

Need Extra Help on these Topics?

The following is a webpage that can assist you in the topics that were covered on this page:

 http://www.sosmath.com/algebra/inequalities/ineq01/ineq01.html This website helps you with linear inequalities.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

Last revised on July 27, 2011 by Kim Seward.