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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?
Chapter 1: Tools of Algebra1-4: 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) 6 + 10 – 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) 6 + 10 – 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) 6 + 10 – 5x < 41 (combine like terms) 16 – 5x < 41 (subtract 16 from each side)
-16 -16 -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) 6 + 10 – 5x < 41 (combine like terms) 16 – 5x < 41 (subtract 16 from each side)
-16 -16 -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 < 3
+ 12 +12 (add 12 to both sides) 3x < 15
1-4: Solving Inequalities Graphing Inequalities
Example: 3x – 12 < 3
+ 12 +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
0 1 2 3 4 5 6 70–1–2–3
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 -2x (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 -2x (subtract 2x from both sides) -3 > -10
(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 $200 + 25% 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 $200 + 25% of ticket sales > $500 Let x = ticket sales (in dollars)
Write an equation and solve 200 + 0.25x > 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 $200 + 25% of ticket sales > $500 Let x = ticket sales (in dollars)
Write an equation and solve 200 + 0.25x > 500
-200 -200 (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 $200 + 25% of ticket sales > $500 Let x = ticket sales (in dollars)
Write an equation and solve 200 + 0.25x > 500
-200 -200 (subtract 200 from each side) 0.25x > 300
0.25 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
+1 +13x > -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
+1 +13x > -273 3x > -9
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
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 +13x > -273 3x > -9
- 7 - 72x < 12
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
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 +13x > -273 3x > -9
- 7 - 72x < 122 2x < 6
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
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 +13x > -273 3x > -9
- 7 - 72x < 122 2x < 6
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
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
+2 +24y > 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
+2 +24y > 164 4y > 4
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
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
+2 +24y > 164 4y > 4
+4 +43y < -9
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
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
+2 +24y > 164 4y > 4
+4 +43y < -93 3y < -3
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
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
+2 +24y > 164 4y > 4
+4 +43y < -93 3y < -3
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
0 1 2 3 4 5 6 7 8 9 10 11 120–1–2–3–4–5–6–7–8–9–10–11
1-4: Solving Inequalities Real World Connection
The ideal length of a bolt is 13.48 cm. The length can vary from the ideal length by at most 0.03 cm. A machinist finds one bolt that is 13.67 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 13.48 cm. The length can vary from the ideal length by at most 0.03 cm. A machinist finds one bolt that is 13.67 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: 13.48 – 0.03 = 13.45 cm Maximum length: Length after cut:
1-4: Solving Inequalities Real World Connection
The ideal length of a bolt is 13.48 cm. The length can vary from the ideal length by at most 0.03 cm. A machinist finds one bolt that is 13.67 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: 13.48 – 0.03 = 13.45 cm Maximum length: 13.48 + 0.03 = 13.51 cm Length after cut:
1-4: Solving Inequalities Real World Connection
The ideal length of a bolt is 13.48 cm. The length can vary from the ideal length by at most 0.03 cm. A machinist finds one bolt that is 13.67 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: 13.48 – 0.03 = 13.45 cm Maximum length: 13.48 + 0.03 = 13.51 cm Length after cut: 13.67 – x
1-4: Solving Inequalities Solution:
Minimum length < length after cut < maximum length
Let x be the amount cut from the bolt Minimum length: 13.48 – 0.03 = 13.45 cm Maximum length: 13.48 + 0.03 = 13.51 cm Length after cut: 13.67 – x
13.45 < 13.67 – 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: 13.48 – 0.03 = 13.45 cm Maximum length: 13.48 + 0.03 = 13.51 cm Length after cut: 13.67 – x
13.45 < 13.67 – x < 13.51-13.67 -13.67 -13.67 (subtract 13.67 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: 13.48 – 0.03 = 13.45 cm Maximum length: 13.48 + 0.03 = 13.51 cm Length after cut: 13.67 – x
13.45 < 13.67 – x < 13.51-13.67 -13.67 -13.67 (subtract 13.67 from all parts)
-0.22 < -x < -0.16 -1 -1 -1 (divide all parts by -1)
0.22 > x > 0.16 (and flip all signs)
1-4: Solving Inequalities Assignment
Page 29 Problems 1 – 27 (odd problems)
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