# Chapter 1: Tools of Algebra 1-4: Solving Inequalities

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Chapter 1: Tools of Algebra 1-4: Solving Inequalities
Essential Question: What is one important difference between solving equations and solving inequalities?

1-4: Solving Inequalities
Inequalities are solved exactly the same as equations except for one key difference: When multiplying or dividing by a negative number, you must reverse the inequality Example: 6 + 5 (2 – x) < 41

1-4: Solving Inequalities
Inequalities are solved exactly the same as equations except for one key difference: When multiplying or dividing by a negative number, you must reverse the inequality Example: 6 + 5 (2 – x) < 41 (distribute) – 5x < 41

1-4: Solving Inequalities
Inequalities are solved exactly the same as equations except for one key difference: When multiplying or dividing by a negative number, you must reverse the inequality Example: 6 + 5 (2 – x) < 41 (distribute) – 5x < 41 (combine like terms) 16 – 5x < 41

1-4: Solving Inequalities
Inequalities are solved exactly the same as equations except for one key difference: When multiplying or dividing by a negative number, you must reverse the inequality Example: 6 + 5 (2 – x) < 41 (distribute) – 5x < 41 (combine like terms) 16 – 5x < 41 (subtract 16 from each side) -5x < 25

1-4: Solving Inequalities
Inequalities are solved exactly the same as equations except for one key difference: When multiplying or dividing by a negative number, you must reverse the inequality Example: 6 + 5 (2 – x) < 41 (distribute) – 5x < 41 (combine like terms) 16 – 5x < 41 (subtract 16 from each side) -5x < 25 (divide both sides by -5) -5 -5 (and flip the sign) x > -5

1-4: Solving Inequalities
Graphing Inequalities Regular equations were graphed simply by putting a point on a number line. Because inequalities imply an infinite number of solutions, we graph them using a line When the variable comes first, you can follow the arrow If the inequality uses < or >, use an open circle If the inequality uses < or >, use a closed circle Think: If you do the extra work and underline the inequality, you have to do the extra work and fill in the circle.

1-4: Solving Inequalities
Graphing Inequalities Example: 3x – 12 < 3

1-4: Solving Inequalities
Graphing Inequalities Example: 3x – 12 < (add 12 to both sides) 3x < 15

1-4: Solving Inequalities
Graphing Inequalities Example: 3x – 12 < (add 12 to both sides) 3x < 15 3 3 (divide both sides by 3) x < 5 x comes first, which means: Put an open circle at 5 (because the inequality is “<“) Draw an arrow to the left

1-4: Solving Inequalities
“No Solutions” or “All Real Numbers” as solutions If all variables get eliminated in a problem, it means that the solution is either “No Solution” or “All Real Numbers” If the statement is false, there is “No Solution” If the statement is true, “All Real Numbers” will solve Example: 2x – 3 > 2(x – 5)

1-4: Solving Inequalities
“No Solutions” or “All Real Numbers” as solutions If all variables get eliminated in a problem, it means that the solution is either “No Solution” or “All Real Numbers” If the statement is false, there is “No Solution” If the statement is true, “All Real Numbers” will solve Example: 2x – 3 > 2(x – 5) 2x – 3 > 2x – 10 (distribute)

1-4: Solving Inequalities
“No Solutions” or “All Real Numbers” as solutions If all variables get eliminated in a problem, it means that the solution is either “No Solution” or “All Real Numbers” If the statement is false, there is “No Solution” If the statement is true, “All Real Numbers” will solve Example: 2x – 3 > 2(x – 5) 2x – 3 > 2x – 10 (distribute) -2x x (subtract 2x from both sides) -3 > -10

1-4: Solving Inequalities
“No Solutions” or “All Real Numbers” as solutions If all variables get eliminated in a problem, it means that the solution is either “No Solution” or “All Real Numbers” If the statement is false, there is “No Solution” If the statement is true, “All Real Numbers” will solve Example: 2x – 3 > 2(x – 5) 2x – 3 > 2x – 10 (distribute) -2x x (subtract 2x from both sides) -3 > (note: you could add 3 to both sides to see 0 > -7) -3 is greater than -10, so “All Real Numbers” are solutions

1-4: Solving Inequalities
Real World Connection Example: A band agrees to play for \$200 plus 25% of the ticket sales. Find the ticket sales needed for the band to receive at least \$500 Cut the meat out of the problem \$ % of ticket sales > \$500 Let x = ticket sales (in dollars) Write an equation and solve

1-4: Solving Inequalities
Real World Connection Example: A band agrees to play for \$200 plus 25% of the ticket sales. Find the ticket sales needed for the band to receive at least \$500 Cut the meat out of the problem \$ % of ticket sales > \$500 Let x = ticket sales (in dollars) Write an equation and solve x > 500

1-4: Solving Inequalities
Real World Connection Example: A band agrees to play for \$200 plus 25% of the ticket sales. Find the ticket sales needed for the band to receive at least \$500 Cut the meat out of the problem \$ % of ticket sales > \$500 Let x = ticket sales (in dollars) Write an equation and solve x > (subtract 200 from each side) 0.25x > 300

1-4: Solving Inequalities
Real World Connection Example: A band agrees to play for \$200 plus 25% of the ticket sales. Find the ticket sales needed for the band to receive at least \$500 Cut the meat out of the problem \$ % of ticket sales > \$500 Let x = ticket sales (in dollars) Write an equation and solve x > (subtract 200 from each side) 0.25x > 300  0.25 (divide each side by 0.25) x > 1200

1-4: Solving Inequalities
Compound Inequality A pair of inequalities combined using the words and or or. Solve the two inequalities separately Inequalities that use “and” are going to meet in the middle They have two ends Inequalities that use “or” are going to go in opposite directions Like oars on a boat

1-4: Solving Inequalities
Example: Compound Inequality using “And” Graph the solution of 3x – 1 > -28 and 2x + 7 < 19 3x – 1 > -28 2x + 7 < 19

1-4: Solving Inequalities
Example: Compound Inequality using “And” Graph the solution of 3x – 1 > -28 and 2x + 7 < 19 3x – 1 > -28 2x + 7 < 19 3x > -27

1-4: Solving Inequalities
Example: Compound Inequality using “And” Graph the solution of 3x – 1 > -28 and 2x + 7 < 19 3x – 1 > -28 2x + 7 < 19 3x > -27 3 3 x > -9

1-4: Solving Inequalities
Example: Compound Inequality using “And” Graph the solution of 3x – 1 > -28 and 2x + 7 < 19 3x – 1 > -28 2x + 7 < 19 3x > -27 3 3 x > -9 2x < 12

1-4: Solving Inequalities
Example: Compound Inequality using “And” Graph the solution of 3x – 1 > -28 and 2x + 7 < 19 3x – 1 > -28 2x + 7 < 19 3x > -27 3 3 x > -9 2x < 12 2 2 x < 6

1-4: Solving Inequalities
Example: Compound Inequality using “And” Graph the solution of 3x – 1 > -28 and 2x + 7 < 19 3x – 1 > -28 2x + 7 < 19 3x > -27 3 3 x > -9 2x < 12 2 2 x < 6

1-4: Solving Inequalities
Example: Compound Inequality using “Or” Graph the solution of 4y – 2 > 14 or 3y – 4 < -13 4y – 2 > 14 3y – 4 < -13

1-4: Solving Inequalities
Example: Compound Inequality using “Or” Graph the solution of 4y – 2 > 14 or 3y – 4 < -13 4y – 2 > 14 3y – 4 < -13 4y > 16

1-4: Solving Inequalities
Example: Compound Inequality using “Or” Graph the solution of 4y – 2 > 14 or 3y – 4 < -13 4y – 2 > 14 3y – 4 < -13 4y > 16 4 4 y > 4

1-4: Solving Inequalities
Example: Compound Inequality using “Or” Graph the solution of 4y – 2 > 14 or 3y – 4 < -13 4y – 2 > 14 3y – 4 < -13 4y > 16 4 4 y > 4 3y < -9

1-4: Solving Inequalities
Example: Compound Inequality using “Or” Graph the solution of 4y – 2 > 14 or 3y – 4 < -13 4y – 2 > 14 3y – 4 < -13 4y > 16 4 4 y > 4 3y < -9 3 3 y < -3

1-4: Solving Inequalities
Example: Compound Inequality using “Or” Graph the solution of 4y – 2 > 14 or 3y – 4 < -13 4y – 2 > 14 3y – 4 < -13 4y > 16 4 4 y > 4 3y < -9 3 3 y < -3

1-4: Solving Inequalities
Real World Connection The ideal length of a bolt is cm. The length can vary from the ideal length by at most 0.03 cm. A machinist finds one bolt that is cm long. By how much should the machinist decrease the length so the bolt can be used? Solution: Minimum length < length after cut < maximum length Let x be the amount cut from the bolt Minimum length: Maximum length: Length after cut:

1-4: Solving Inequalities
Real World Connection The ideal length of a bolt is cm. The length can vary from the ideal length by at most 0.03 cm. A machinist finds one bolt that is cm long. By how much should the machinist decrease the length so the bolt can be used? Solution: Minimum length < length after cut < maximum length Let x be the amount cut from the bolt Minimum length: – 0.03 = cm Maximum length: Length after cut:

1-4: Solving Inequalities
Real World Connection The ideal length of a bolt is cm. The length can vary from the ideal length by at most 0.03 cm. A machinist finds one bolt that is cm long. By how much should the machinist decrease the length so the bolt can be used? Solution: Minimum length < length after cut < maximum length Let x be the amount cut from the bolt Minimum length: – 0.03 = cm Maximum length: = cm Length after cut:

1-4: Solving Inequalities
Real World Connection The ideal length of a bolt is cm. The length can vary from the ideal length by at most 0.03 cm. A machinist finds one bolt that is cm long. By how much should the machinist decrease the length so the bolt can be used? Solution: Minimum length < length after cut < maximum length Let x be the amount cut from the bolt Minimum length: – 0.03 = cm Maximum length: = cm Length after cut: – x

1-4: Solving Inequalities
Solution: Minimum length < length after cut < maximum length Let x be the amount cut from the bolt Minimum length: – 0.03 = cm Maximum length: = cm Length after cut: – x 13.45 < – x < 13.51

1-4: Solving Inequalities
Solution: Minimum length < length after cut < maximum length Let x be the amount cut from the bolt Minimum length: – 0.03 = cm Maximum length: = cm Length after cut: – x 13.45 < – x < (subtract from all parts) -0.22 < -x < -0.16

1-4: Solving Inequalities
Solution: Minimum length < length after cut < maximum length Let x be the amount cut from the bolt Minimum length: – 0.03 = cm Maximum length: = cm Length after cut: – x 13.45 < – x < (subtract from all parts) -0.22 < -x <   -1  (divide all parts by -1) 0.22 > x > (and flip all signs)

1-4: Solving Inequalities
Assignment Page 29 Problems 1 – 27 (odd problems)