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 Intermediate Algebra Tutorial 10: Linear Inequalities

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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. Write the answer to an inequality using interval notation. 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.  I think you are ready to get going on this tutorial.

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

 Interval Notation

Interval notation is a way to notate the range of values that would make an inequality true.  There are two types of intervals, open and closed (described below), each with a specific way to notate it so we can tell the difference between the two.

Note that in the interval notations (found below), you will see the symbol  , which means infinity

 Positive infinity () means it goes on and on indefinitely to the right of the number - there is no endpoint on the right.  Negative infinity (-) means it goes on and on indefinitely to the left of the number - there is no endpoint to the left.  Since we don't know what the largest or smallest numbers are, we need to use infinity or negative infinity to indicate there is no endpoint in one direction or the other.

In general, when using interval notation, you always put the smaller value of the interval first (on the left side), put a comma between the two ends, then put the larger value of the interval (on the right side).  You will either use a curved end ( or ) or a boxed end [ or  ], depending on the type of interval (described below).

If you have either infinity or negative infinity on either end, you always use a curve for that end. This will indicate that there is no definite endpoint in that direction,  it keeps going and going.

 Open Interval

 An open interval does not include where your variable is equal to the endpoint. To indicate this, we use a curved end as shown below.

 Inequality x > a  x < a  x > 4 x < 4 Interval Notation for Open Intervals (a, ) (-, a) (4, ) (-, 4)

 Closed Interval

 A closed interval includes where your variable is equal to the endpoint. To indicate this, we use a boxed end as shown below.  As mentioned above, even though a is included and has a boxed end, if it goes to either infinity or negative infinity on the other end, we will notate it with a curved end for that end only!

 Inequality x > a  x < a  x > 4 x < 4 Interval Notation for Closed Intervals [a, ) (-, a] [4, ) (-, 4]

 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 1:  Solve, write your answer in interval notation and graph the solution set.

 Interval notation: (-, 4) Graph: *Inv. of sub. 7 is add. 7   *Open interval indicating all values less than 4 *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. Interval notation: We have an open interval since we are not including where it is equal to 4.   x is less than  4, so 4 is the largest value of the interval, so it goes on the right.  Since there is no lower endpoint (it is ALL values less than 4), we put the negative infinity symbol on the left side.  The curved end on 4 indicates an open interval.  Negative infinity always has a curved end because there is not an endpoint on that side. Graph:  We use the same type of notation on the endpoint as we did in the interval notation,  a curved end.  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.

 Example 2: Solve, write your answer in interval notation and graph the solution set.

 Interval notation: [-5, ) Graph: *Inv. of add 10 is sub. 10   *Closed interval indicating all values greater than or = -5 *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. Interval notation: We have a closed interval since there we are including where it is equal to -5.   x is greater than or equal to -5, so -5 is our smallest value of the interval, so it goes on the left.  Since there is no upper endpoint (it is ALL values greater than or equal to -5), we put the infinity symbol on the right side.  The boxed end on -5 indicates a closed interval.  Infinity always has a curved end because there is not an endpoint on that side. Graph:  We use the same type of notation on the endpoint as we did in the interval notation,  a  boxed end.  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.

 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 3:  Solve, write your answer in interval notation and graph the solution set.

 Interval notation: (-, -2) Graph: *Inv. of mult. by 5 is div. by 5     *Open interval indicating all values less than -2 *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. Interval notation: We have an open interval since there we are not including where it is equal to -2.   x is less than  -2, so -2 is our largest value of the interval, so it goes on the right.  Since there is no lower endpoint (it is ALL values less than -2), we put the negative infinity symbol on the left side.  The curved end on -2 indicates an open interval.  Negative infinity always has a curved end because there is not an endpoint on that side. Graph:  We use the same type of notation on the endpoint as we did in the interval notation,  a curved end.  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.

 Example 4:  Solve, write your answer in interval notation and graph the solution set.

 Interval notation: (3, ) Graph: *Inv. of div. by 3 is mult. by 3     *Open interval indicating all values greater than 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. Interval notation: We have an open interval since there we are not including where it is equal to 3.   x is greater than 3, so 3 is our smallest value of the interval so it goes on the left.  Since there is no upper endpoint (it is ALL values less than 3) we put the infinity symbol on the right side.  The curved end on 3 indicates an open interval.  Infinity always has a curved end because there is not an endpoint on that side. Graph:  We use the same type of notation on the endpoint as we did in the interval notation,  a curved end.  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.

 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 5:  Solve, write your answer in interval notation and graph the solution set.

 Interval notation: (- , -14) Graph: *Inv. of div. by -2 is mult. by -2, so reverse inequality sign     *Open interval indicating all values less than -14 *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. Interval notation: We have an open interval since there we are not including where it is equal to -14.   x is less than  -14, so -14 is our largest value of the interval, so it goes on the right.  Since there is no lower endpoint (it is ALL values less than -14), we put the negative infinity symbol on the left side.  The curved end on -14 indicates an open interval.  Negative infinity always has a curved end because there is not an endpoint on that side. Graph:  We use the same type of notation on the endpoint as we did in the interval notation,  a curved end.  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.

 Example 6:  Solve, write your answer in interval notation and graph the solution set.

 Interval notation: [-3, ) Graph: *Inv. of mult. by -3 is div. by -3, so reverse inequality sign     *Closed interval indicating all values greater than or = -3 *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. Interval notation: We have a closed interval since there we are including where it is equal to -3.   x is greater than or equal to -3, so -3 is our smallest value of the interval so it goes on the left.  Since there is no upper endpoint (it is ALL values greater than or equal to -3), we put the infinity symbol on the right side.  The boxed end on -3 indicates a closed interval.  Infinity always has a curved end because there is not an endpoint on that side. Graph:  We use the same type of notation on the endpoint as we did in the interval notation,  a  boxed end.  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.

 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 7: Linear Equations in One Variable.

Example 7: Solve, write your answer in interval notation and graph the solution set.

 Interval notation: (-3, ) Graph: *Inv. of sub. 3 is add. 3   *Inv. of mult. by -3 is div. both sides by -3, so reverse inequality sign *Open interval indicating all values greater than -3   *Visual showing all numbers greater than -3 on the number line

 Interval notation: We have an open interval since there we are not including where it is equal to -3.   x is greater than -3, so -3 is our smallest value of the interval so it goes on the left.  Since there is no upper endpoint (it is ALL values less than -3), we put the infinity symbol on the right side.  The curved end on -3 indicates an open interval.  Infinity always has a curved end because there is not an endpoint on that side. Graph:  We use the same type of notation on the endpoint as we did in the interval notation,  a curved end.  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.

 Example 8:  Solve, write your answer in interval notation and graph the solution set.

 Interval notation: (-, -1/2) Graph: *Distributive property *Get x terms on one side, constants on the other side *Inv. of mult. by 2 is div. by 2       *Open interval indicating all values less than -1/2   *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.    Interval notation: Again, we have an open interval since we are not including where it is equal to -1/2.   This time  x is less than  -1/2, so -1/2 is our largest value of the interval so it goes on the right.  Since there is no lower endpoint (it is ALL values less than -1/2), we put the negative infinity symbol on the left side.  The curved end on -1/2 indicates an open interval.  Negative infinity always has a curved end because there is not an endpoint on that side. Graph:  Again, we use the same type of notation on the endpoint as we did in the interval notation, a curved end.  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.

 Example 9:  Solve, write your answer in interval notation and graph the solution set.

 Interval notation: (-, 4] 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   *Closed interval indicating all values less than or equal to 4   *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.  Interval notation: This time we have a closed interval since we are including where it is equal to 4.   x is less than or equal to  4, so 4 is our largest value of the interval so it goes on the right.  Since there is no lower endpoint (it is ALL values less than or equal to 4), we put the negative infinity symbol on the left side.  The boxed end on 4 indicates a closed interval.  Negative infinity always has a curved end because there is not an endpoint on that side. Graph:  Again, we use the same type of notation on the endpoint as we did in the interval notation,  a boxed end this time.  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.

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, write your answer in interval notation and graph the solution set.

Need Extra Help on these Topics?

The following are webpages 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. http://www.math.com/school/subject2/lessons/S2U3L4DP.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 3, 2011 by Kim Seward.
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