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### Solving and graphing inequalities lecture 1

• 1. Solving and graphing inequalities UNIT TOPIC : REASONING WITH EQUATIONS AND INEQUALITIES CLASS: ALGEBRA 2 FOR 9TH-12TH GRADE PRESENTED BY MS. ZAMBRANO
• 2. Student Objectives  Understand the properties of inequalities  Learn how to solve singular inequality expressions  Learn how to solve absolute value inequalities  Identify how to graph inequalities on a number line  Implement the use of inequalities to real life situations CA standards that will be met  Compare models  Graphing of functions
• 3. How do inequalities play into effect in our daily lives ?  Have any of you seen this type of stop sign on your way to or from school? What does it represent?  As our general knowledge we know that the sign represent the hours we are able to park  We know that we can park between the hours of 9 AM and 7 PM (19 hour)  This can be represented by the inequality 9 ≤ 𝑥 ≤ 19
• 4. Fill in the KWL Chart before we start lecture What I know: What I want to learn: What I learned:  As a guide for our lecture let us fill in the first two columns of our KWL chart  You will take 5 min to fill in anything you know about inequalities and what you would like to learn about inequalities
• 5. Key information for inequalities  An inequality makes a compares between two values.  there are four different compares that can be made and represented 1) < - represents the less than sign, and can be depicted with an open dot 2) > - represents the greater than sign, and can be depicted with an open dot 3) ≥ - represents greater than or equal to sign, and can be depicted with a closed dot 4) ≤ - represents less then or equal to, and can be depicted with a close dot  Absolute value - can be written as |a| or ±𝑎 which means that the value of a is both a positive and negative. Where a is represented by any positive number.  If the value of a is negative the a automatically becomes a positive number since the absolute value neutralizes the a.
• 6. Properties of inequalities video  As we watch the following video fill in the boxes provided in the lecture handout  Note the 3 important properties of inequalities.  Note the a,b, and c represent whole numbers which are numbers that range from any negative number to any positive number
• 7. Steps in solving inequalities 1)Determine what type of inequality you have simple comparison or an absolute value inequality 2) Solve for the specific variable. If given a comparison directly solve the equation if given an absolute, we must show that a contains both a positive and negative number that the inequality is set equal to. (a < x < -a) 3) When solving inequalities we must remember that whatever we do to one side we must do to the other. (This include multiplying, adding, subtracting, and dividing) 4) Graph the inequalities based on the comparison inequality we found.
• 8. Example problem #1 5x -2 < 3 5x -2+2 < 3+2 5x < 5 5 5 x < 5 5 X < 1 2 -1 0 1 2 3 4  Determine that it is a regular inequality equation  Add 2 to both sides and divide by 5 to both sides to isolate the x variable  Obtain the new inequality x< 1  Since the inequality is a less then inequality, we use an open circle.  Since the inequality is not bounded from the negative side it is continuous
• 9. Example problem #2 solving absolute inequalities |4x-8|≤ 4 -4 ≤ 4x-8 ≤ 4 -4 + 8 ≤ 4x ≤ 4 + 8 4 ≤ 4x ≤ 12 4 4 ≤ x ≤ 12 4 1 ≤ x ≤ 3 -2 -1 0 1 2 3 4  Rewrite the problem since it has an absolute value. We obtain both a positive and negative value of 4  Use algebra to solve for x. We add the 8 to both side and divide by 4 to both sides.  We obtain an inequality 1 ≤ x ≤ 3 which can be graphed  Since the inequality contains a greater than or equal to and a less then and equal to the graph contains closed dots on 1 and 3.
• 10. Now take 10 min to apply what you learned and solve a and b on your own A) 5|2x-4| < 10 B) |-3x-2| > 6
• 11. Answers to problems A and B A) 5|2x-4| < 10 |2x-4| < 2 -2 < 2x-4 < 2 2 < 2x < 6 1 < x < 3 -1 0 1 2 3 4 5 B) |4x-2| ≤ 6 -6≤ 4x -2 ≤ 6 -4 ≤ 4x ≤ 8 -1 ≤ x ≤ 2 -2 -1 0 1 2 3
• 12. Example problem 3 solving inequalities |x+2| > 3 - 9x -3+9x > x+2 > 3-9x -2-3+9x > x > 3-9x -2 -5 +9x > x > 1-9x or -5 + 9x > x x > 1-9x -5 > -8x 10x > 1 5 8 < x x > 1/10 -2 -1 0 1 2 3 4  Since I have an absolute value, I take the positive and negative of the equation 3-9x.  I subtract a 2 to both sides of the inequality  Since in this case both inequalities are facing the same direction and I still need to solve for x I create two separate equations  I solve for x in each of the separate equations using algebra.  Now we obtain 2 answers  Since both inequalities illustrate that x is greater than 5 8 and 1 10 we need to pick the smallest value of the two since x will always be grater then that value.  Our graph will then start at 1 10 since it is smaller than 5/8 and will be continuous since the value of x will always be any number greater than 1/10.
• 13. Example problem 4 2|x-4| ≥ 4 – 8x |x-4| ≥ 2 -4x -2 +4x ≥ x-4 ≥ 2-4x 2 + 4x ≥ x ≥ 6-4x Or 2 + 4x ≥ x x ≥ 6-4x 2 ≥ -3x 5x ≥ 6 2≤3x x ≥ 6 5 2 3 ≤ x -1 0 1 2 3 4  Divide both sides by 2  Set the inequalities equal to the positive and negative version of 2-4x  Separate into 2 equations  Solve for x for the two inequalities  Graph the value of x which is continuous from 1.5
• 14. Now take the time to think and answer these questions with a partner .  How do simple inequalities differ from absolute value inequalities?  How are we able to determine whether the graph of the inequality will have an open or closed dot ?  In the previous problem why was the value of x continuous between [1.5, )
• 15. Real world inequality problem Karla has \$30 and wants to buy cupcakes for her friend Jade but does not want to spend more than \$20. Recently she found a bakery website that sells cupcakes at a price of \$2 for each cupcake with a fee of \$6 for delivery. How many cupcakes can she buy for her friend without going over her spending limit? 1) Determine the important information 2)Create an inequality equation 2x+6 ≤20 2)Solve the inequality equation using algebra 2x ≤ 14 x ≤ 7 3) Interpret the information Karla is only able to buy 7 cupcakes for her friend jade without going over her spending limit.
• 16. Fill in the last column of the KWL chart What I know: What I want to learn: What I learned:
• 17. Last culminating activity After everything you have learned with a partner take the time to analyze this final question and determine which of these two inequalities can be solved? For the one that can be solved graph its inequality. For the one that can not be solved explain why the inequality can not be solved. a)|5x+8| < -3 b)2|4x-2| > 2
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