MATH-A 100: Sec. 2.8 Detailed Notes - Inequalities

Math A100 IUSB Sec. 2.8 Detailed Notes Inequalities “ Inequalities ” are when the answer can be any one of many possibilities. For example, “ you must be 21 or over to purchase alcohol ” . So, any age 21 or more can do so. Instead of writing out all the possibilities, we can represent the “ set ” of answers using a <, >, <, or > sign. Recall that: < - is called the ” Less than ” sign, i.e. the number of the left is smaller than the one on the right > - is called the ” Greater than ” sign, i.e. the number of the left is BIGGER than the one on the right < - is called the ” Less than or equal to ” sign, i.e. the number of the left is the same as or smaller than the one on the right > - is called the ” Greater than or equal to ” sign, i.e. the number of the left is the same as or BIGGER than the one on the right So, typically, you would write the answer to example above: x > 21 Since the answer is a set, or a range, of numbers, we have FOUR ways we can represent the answer: 1) Using Inequality notation : x > 21 2) Graphing the answer on the number line 3) Using Set Builder Notation 4) Using Interval Notation Graphing inequalities Graphing inequalities is done on a number line very similar to graphing regular solutions. The big difference is that the “ solution ” (i.e. the answer) is more than one number, we have to show that on the number line. To do this, we use arrows (  ) point in whichever direction (left or right) that the numbers continue in. Using the example above: x > 21 a. Find your answer on the number line b. Decide whether your answer is included or not a. If it IS included, i.e. your answer uses the < or > sign, we will use the brackets, either [ or ] instead of the dots or filled in circles - MML has no way to put dots in, so they switched to using [ & ] b. If your answer is NOT included, i.e. your answer uses the < or > sign, we will use the brackets, either ( or ) instead of the open circles - again MML has no way to put open circle in, so they switched to using ( & )

c. Decide which direction your arrow should go and draw it in So, graphing the example above, x > 21, would look like: [ | 21 NOTE: I posted some graphing paper in the Useful Resources File in Canvas that you can download and print off. Set-Builder Notation Set-Builder notation is a way of writing the answer as a set of values. If you remember from back in elementary ` or maybe middle school, we used the braces, { and }, to represent “ sets ” . We will use them here as well. The “ basic ” form for Set-builder notation is: x = {x | _____} where what follows the “ | ” is the inequality notation (see above). So for this example above, when x > 21, then Set-Builder notation would be: In Set-Builder Notation: The solution set = {x | x > 21 } This is read: “ the solution set is ‘ x ’ such that x > 21 ” or “ the solution set is ‘ x ’ when that x > 21 ” All Set-Builder notation really does is show the answers as a set of values, instead of as a drawing (a graph). Interval Notation “ Interval Notation ” is yet another way to write the answer, only here we do not use any variables or any of the four inequality signs. “ Interval Notation ” is designed to show you what numbers are included the answer by showing you the “ interval ” , or range of numbers, the answer can come from. Interval Notation looks a lot like ordered pairs (coordinates of a point), but use [ and ] in addition to ( and ). Interval notation simply shows the left-most number an the right-most number, with a “ comma ” in between to separate them. Just like in graphing inequalities, if the number IS included, use either [ or ] and if the number is NOT included, then use ( or ). For the example above, the left most number would be 21, since no values smaller than 21 would work. The arrow points to the right and goes on forever, so the right-most number would be infinity, i.e. ∞ . So, in Interval Notation: The solution is [21, ∞) NOTE: since we do not have an actual value for infinity, . ∞, or negative infinity, -. ∞, then we use the parentheses, ( or ), instead of the brackets, [ or ]. ALSO NOTE: There is shortcut/hint for writing the Interval Notation  you can take it directly from the graph of the inequality: whatever your “ endpoints ” are on the graph are also the left-most & right-most values used in the Interval Notation AND whatever symbols you used on the graph, [ or ] or ( or ), you also use in the Interval Notaton.

Rewriting Inequalities In simpler form You will be asked in the HW to re-write an inequality in simpler way. For example: x + 5 < 12 implies that x < _____ where they want you to fill in the blank. “ x + 5 < 12 ” means that x must be smaller than 7, so your answer would be: x < 7 Similarly, 4x > 12 implies that x > ____ again they want you to fill in the blank. “ 4x > 12 ” means that x must be bigger than 3, so your answer would be: x > 3 Another way you may see these same examples is: x + 5 < 12 implies that x __ 7 where they want you to fill in the blank with an inequality sign “ x + 5 < 12 ” means that x must be smaller t han 7, so your answer would be: x < 7 4x > 12 implies that x __ 3 again they want you to fill in the blank with an inequality sign “ 4x > 12 ” means that x must be bigger than 3, so your answer would be: x > 3 Solving Inequalities The next logical step would be to have you solve an inequality and list your answer/solution in one or all of the four notations. To solve an inequality, you would use the same techniques you use in solving regular equations: a. Undo what they gave you b. make it into a Proportion & cross-multiply c. Clear the fractions d. Etc. Checking your answer : Since your answer/solution can be any one of many values, to check your answer  simply pick any number in the range of your answer/solution and put it back in your original inequality to check your work. Example 1: Solve and write your answer in the four notations. 3x – 7 < 11 Solving: add 7 to both sides, giving you: 3x <18

Divide both sides by 3, giving you: x < 6 Four notations: 1) Using Inequality notation : x < 6 2) Graphing the answer on the number line ) | 6 3) Using Set Builder Notation : { x | x < 6 } 4) Using Interval Notation: ( - ∞ , 6) NOTE: We used the “ ) ” with the 6 because it is not included, i.e. if x = 6, you get 6 < 6, which is not true. ALSO NOTE : The ∞ symbol in MML can be found in the toolbar that shows up at the bottom of the HW question page once you click inside the answer box. You will have to type the “ - “ with the infinity symbol to get negative infinity  MML does not offer a “ - ∞” symbol in the toolbar. Checking your work: since x < 6, any number less than 6 should work. So, let ’ s put in “ 0 ” for “ x ” and see if your work is correct ( I like to use “ 0 ” because it simplifies most of the math work ): 3x – 7 < 11 becomes 3(0) – 7 < 11  0 – 7 < 11  -7 < 11  TRUE , so your work checks ! NOTE : Another way to check is to pick a number that is NOT in the “ solution range ” and put it in the original inequality . You should get a FALSE answer. For Example 1: x < 6  “ 10 ” is NOT in the answer range, so put in “ 10 ” for “ x ” 3x – 7 < 11 becomes 3(10) – 7 < 11  30 – 7 < 11  23 < 11  FALSE , as you would expect so your work checks ! Example 2: Solve and write your answer in the four notations. 3x – 7 > x – 15 Solving: get your “ x ” s together on one side, giving you: 2x – 7 > – 15 add 7 to both sides, giving you: 2x > - 8

Divide both sides by 3, giving you: x > -4 Four notations: 1) Using Inequality notation : x > -4 2) Graphing the answer on the number line ( | ‘ - 4 3) Using Set Builder Notation : { x | x > - 4 } 4) Using Interval Notation: ( - 4, ∞ ) Checking your work: since x > - 4, any number bigger than - 4 should work. So, let ’ s put in “ 0 ” for “ x ” and see if your work is correct ( I like to use “ 0 ” because it simplifies most of the math work ): 3x – 7 > x – 15 becomes 3(0) – 7 < (0) - 15  0 – 7 < 0 - 11  -7 < -15  TRUE Example 3: Solve and write your answer in the four notations. 3x – 7 < 5x – 15 Solving: get your “ x ” s together on one side, giving you: – 7 < 2x – 15 add 15 to both sides, giving you: 8 > 2x Divide both sides by 3, giving you: 4 > x Four notations: 1) Using Inequality notation : x < 4 NOTE : you always write the “ x ” first in your answer, so you may have to reverse the order to write it correctly 2) Graphing the answer on the number line ] | 4

3) Using Set Builder Notation : { x | x < 4 } 4) Using Interval Notation: ( - ∞, 4] Checking your work: since x < 4, any number less than 4 should work. So, let ’ s put in “ 0 ” for “ x ” and see if your work is correct ( I like to use “ 0 ” because it simplifies most of the math work ): 3x – 7 < 5x – 15 becomes 3(0) – 7 < (0) - 15  0 – 7 < 0 - 15  -7 < -15  TRUE Special Cases Special Case #1 - keep an eye out for when you multiply or divide by a negative number, i.e. when the number in front of “ x ” is negative. You have two options to choose from: 1) Memorize “ the rule ” 2) Get rid of the negative sign Option 1 - The “ rule ” to memorize: The “ rule ” says that if you multiply or divide by a negative number , you have to flip/reverse the direction your inequality is pointing. Example 1: Solve -3x + 7 > 13 Solving: Subtract 7 from both sides, giving you: -3x + 7 > 13 becomes -3x > 6 Divide both sides by -3 and flip the direction of the >: -3x > 6 becomes x < -2 NOTE : many students mistakenly use this “ rule ” whenever they see a “ negative ” number , so you have to remember exactly when you can use it! For example: 3x - 7 > -13 Solving: Add 7 from both sides, giving you: 3x - 7 > - 13 becomes 3x > - 6 Divide both sides by 3 3 and flip the direction of the >: 3x > - 6 becomes x < -2 Checking : since this student came up with x <-2, then x = -5 should work  let ’ s try it: 3(-5) – 5 > 13  -15- 7 > 13  -22 > 13 ??????? FALSE Option 2 – Getting rid of the negative ( the common sense approach) Simply move your variable to the other side of the “ = ” sign to make the number out front positive  then you don ’ t have to worry about any “ rule to remember!

Example 1: Solve -3x + 7 > 13 Solving: Move the -3x to the other side of the equation by adding 3x to both sides, giving you: -3x + 7 > 13 becomes 7 > 3x + 13 Subtract 13 from both sides, giving you: 7 > 3x + 13 becomes - 6 > 3x Divide both sides by 3, giving you: - 6 > 3x becomes -2 > x which is the same as x < - 2 as above Special Case #2 - No solution Just like with regular equations, you can have inequalities that cannot have a solution. The key here is to focus on the inequality itself. Example 1: 6x - 3(2x – 5) > 18 6x - 3(2x – 5) > 18 distribute to break up the ( ), giving you 6x - 6x + 15 > 18 put your like terms together to get 15 > 18 NOT true  No Solution Example 2: 6x - (2x – 5) < - (7 – 3x) + x 6x - (2x – 5) < - (7 – 3x) + x distribute to break up the ( ), giving you 6x - 2x + 5 < - 7 + 3x + x put your like terms together to get 4x + 5 < -7 + 4x subtract 4x to undo the positive 4x to get 5 < -7 NOT true  No Solution Example 3: 6x - (2x + 5) < -5(1 – x) - x 6x - (2x + 5) < -5(1 – x) - x distribute to break up the ( ), giving you 6x - 2x - 5 < -5 + 5x - x put your like terms together to get 4x - 5 < -5 + 4x subtract 4x to undo the positive x to get - 5 < -5 NOT true  No Solution Special Case #3 - All real numbers You can have inequalities whose answers are “ All Real Solutions ” . The key here again is to focus on the inequality itself. Note: these are the same example as just above, except the inequality is different. Example 1: 6x - 3(2x – 5) < 18 6x - 3(2x – 5) < 18 distribute to break up the ( ), giving you

6x - 6x + 15 < 18 put your like terms together to get 15 < 18 ALWAYS TRUE  All Real Numbers Example 2: 6x - (2x – 5) > - (7 – 3x) + x 6x - (2x – 5) > - (7 – 3x) + x distribute to break up the ( ), giving you 6x - 2x + 5 > - 7 + 3x + x put your like terms together to get 4x + 5 > -7 + 4x subtract 4x to undo the positive 4x to get 5 > -7 NOT true  No Solution Example 3: 6x - (2x + 5) < -5(1 – x) - x 6x - (2x + 5) < -5(1 – x) - x distribute to break up the ( ), giving you 6x - 2x - 5 < -5 + 5x - x put your like terms together to get 4x - 5 < -5 + 4x subtract 4x to undo the positive x to get - 5 < -5 always true  All Real Numbers Videos that may help Solve Linear Inequalities (Interval notation) http://www.youtube.com/watch?v=SXrxEmmfp7A Compound Inequalities (interval notation) http://www.youtube.com/watch?v=o5clgmyYCZE Calculate grade needed on final exam to get given grade using weighted averages http://www.youtube.com/watch?v=UjXbLWCwwg0&feature=youtu.be Calculate grade needed on last test to get certain average grade http://www.youtube.com/watch?v=eEY3U3x_THc