section 4.6 - polynomial inequalities and rational inequalitiesain.faculty.unlv.edu/124...
TRANSCRIPT
![Page 1: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/1.jpg)
Section 4.6
Polynomial Inequalities and Rational Inequalities
![Page 2: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/2.jpg)
Polynomial Inequalities
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Method 1: Graphing
Let’s work through the process using (x + 1)(x + 2)(2x − 5) < 0.
1. Add/subtract terms to get zero on the right-hand-side (RHS) of theinequality.
2. Factor the polynomial on the left-hand-side (LHS) and then graph it.
Leading Coefficient: 2x3
Zeros:
-2 with multiplicity 1
-1 with multiplicity 1
52 = 2.5 with multiplicity 1
−2 −1 1 2 3
![Page 4: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/4.jpg)
Method 1: Graphing
Let’s work through the process using (x + 1)(x + 2)(2x − 5) < 0.
1. Add/subtract terms to get zero on the right-hand-side (RHS) of theinequality.
2. Factor the polynomial on the left-hand-side (LHS) and then graph it.
Leading Coefficient: 2x3
Zeros:
-2 with multiplicity 1
-1 with multiplicity 1
52 = 2.5 with multiplicity 1
−2 −1 1 2 3
![Page 5: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/5.jpg)
Method 1: Graphing
Let’s work through the process using (x + 1)(x + 2)(2x − 5) < 0.
1. Add/subtract terms to get zero on the right-hand-side (RHS) of theinequality.
2. Factor the polynomial on the left-hand-side (LHS) and then graph it.
Leading Coefficient: 2x3
Zeros:
-2 with multiplicity 1
-1 with multiplicity 1
52 = 2.5 with multiplicity 1
−2 −1 1 2 3
![Page 6: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/6.jpg)
Method 1: Graphing
Let’s work through the process using (x + 1)(x + 2)(2x − 5) < 0.
1. Add/subtract terms to get zero on the right-hand-side (RHS) of theinequality.
2. Factor the polynomial on the left-hand-side (LHS) and then graph it.
Leading Coefficient: 2x3
Zeros:
-2 with multiplicity 1
-1 with multiplicity 1
52 = 2.5 with multiplicity 1
−2 −1 1 2 3
![Page 7: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/7.jpg)
Method 1: Graphing
Let’s work through the process using (x + 1)(x + 2)(2x − 5) < 0.
1. Add/subtract terms to get zero on the right-hand-side (RHS) of theinequality.
2. Factor the polynomial on the left-hand-side (LHS) and then graph it.
Leading Coefficient: 2x3
Zeros:
-2 with multiplicity 1
-1 with multiplicity 1
52 = 2.5 with multiplicity 1
−2 −1 1 2 3
![Page 8: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/8.jpg)
Method 1: Graphing (continued)
3. Read the answer off of the graph:
f (x) > 0 is asking wherethe graph is above the x-axis.
f (x) ≥ 0 is asking wherethe graph is at or abovethe x-axis.
f (x) < 0 is asking wherethe graph is below the x-axis.
f (x) ≤ 0 is asking wherethe graph is at or belowthe x-axis.
![Page 9: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/9.jpg)
Method 1: Graphing (continued)
For our example, we’re looking at
(x + 1)(x + 2)(2x − 5) < 0
−2 −1 1 2 3
Answer: (−∞,−2) ∪(−1, 52
)
![Page 10: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/10.jpg)
Method 1: Graphing (continued)
For our example, we’re looking at
(x + 1)(x + 2)(2x − 5) < 0
−2 −1 1 2 3
Answer: (−∞,−2) ∪(−1, 52
)
![Page 11: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/11.jpg)
Method 2: Numerical
Let’s work through the process using x2 − x − 13 ≥ 2x + 5.
1. Add/subtract terms to get zero on the right-hand-side (RHS) of theinequality.
x2 − 3x − 18 ≥ 0
2. Factor the polynomial on the left-hand-side (LHS).
(x − 6)(x + 3) ≥ 0
3. Set the factors on the LHS equal to zero and solve.
x − 6 = 0 or x + 3 = 0x = 6 or x = −3
![Page 12: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/12.jpg)
Method 2: Numerical
Let’s work through the process using x2 − x − 13 ≥ 2x + 5.
1. Add/subtract terms to get zero on the right-hand-side (RHS) of theinequality.
x2 − 3x − 18 ≥ 0
2. Factor the polynomial on the left-hand-side (LHS).
(x − 6)(x + 3) ≥ 0
3. Set the factors on the LHS equal to zero and solve.
x − 6 = 0 or x + 3 = 0x = 6 or x = −3
![Page 13: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/13.jpg)
Method 2: Numerical
Let’s work through the process using x2 − x − 13 ≥ 2x + 5.
1. Add/subtract terms to get zero on the right-hand-side (RHS) of theinequality.
x2 − 3x − 18 ≥ 0
2. Factor the polynomial on the left-hand-side (LHS).
(x − 6)(x + 3) ≥ 0
3. Set the factors on the LHS equal to zero and solve.
x − 6 = 0 or x + 3 = 0x = 6 or x = −3
![Page 14: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/14.jpg)
Method 2: Numerical (continued)
4. Plot the values from step 3 on a number line and pick test pointson each side and between these numbers.
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Test Points
If the inequality is “<” or “>”, plot them with open circlesIf the inequality is “≤” or “≥”, plot them with closed circles
![Page 15: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/15.jpg)
Method 2: Numerical (continued)
4. Plot the values from step 3 on a number line and pick test pointson each side and between these numbers.
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Test Points
If the inequality is “<” or “>”, plot them with open circlesIf the inequality is “≤” or “≥”, plot them with closed circles
![Page 16: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/16.jpg)
Method 2: Numerical (continued)
4. Plot the values from step 3 on a number line and pick test pointson each side and between these numbers.
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Test Points
If the inequality is “<” or “>”, plot them with open circles
If the inequality is “≤” or “≥”, plot them with closed circles
![Page 17: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/17.jpg)
Method 2: Numerical (continued)
4. Plot the values from step 3 on a number line and pick test pointson each side and between these numbers.
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Test Points
If the inequality is “<” or “>”, plot them with open circlesIf the inequality is “≤” or “≥”, plot them with closed circles
![Page 18: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/18.jpg)
Method 2: Numerical (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
(x − 6)(x + 3) ≥ 0
x = −4 : (−4− 6)(−4 + 3) = (−10)(−1) = 10 ≥ 0
x = 0 : (0− 6)(0 + 3) = (−6)(3) = −18 ≥ 0
x = 7 : (7− 6)(7 + 3) = (1)(10) = 10 ≥ 0
6. Fill in the “good” intervals and read the answer off the number line.
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Answer: (−∞,−3] ∪ [6,∞)
![Page 19: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/19.jpg)
Method 2: Numerical (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
(x − 6)(x + 3) ≥ 0
x = −4 : (−4− 6)(−4 + 3) = (−10)(−1) = 10 ≥ 0
x = 0 : (0− 6)(0 + 3) = (−6)(3) = −18 ≥ 0
x = 7 : (7− 6)(7 + 3) = (1)(10) = 10 ≥ 0
6. Fill in the “good” intervals and read the answer off the number line.
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Answer: (−∞,−3] ∪ [6,∞)
![Page 20: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/20.jpg)
Method 2: Numerical (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
(x − 6)(x + 3) ≥ 0
x = −4 : (−4− 6)(−4 + 3) = (−10)(−1) = 10 ≥ 0
x = 0 : (0− 6)(0 + 3) = (−6)(3) = −18 ≥ 0
x = 7 : (7− 6)(7 + 3) = (1)(10) = 10 ≥ 0
6. Fill in the “good” intervals and read the answer off the number line.
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Answer: (−∞,−3] ∪ [6,∞)
![Page 21: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/21.jpg)
Method 2: Numerical (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
(x − 6)(x + 3) ≥ 0
x = −4 : (−4− 6)(−4 + 3) = (−10)(−1) = 10 ≥ 0
x = 0 : (0− 6)(0 + 3) = (−6)(3) = −18 ≥ 0
x = 7 : (7− 6)(7 + 3) = (1)(10) = 10 ≥ 0
6. Fill in the “good” intervals and read the answer off the number line.
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Answer: (−∞,−3] ∪ [6,∞)
![Page 22: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/22.jpg)
Method 2: Numerical (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
(x − 6)(x + 3) ≥ 0
x = −4 : (−4− 6)(−4 + 3) = (−10)(−1) = 10 ≥ 0
x = 0 : (0− 6)(0 + 3) = (−6)(3) = −18 ≥ 0
x = 7 : (7− 6)(7 + 3) = (1)(10) = 10 ≥ 0
6. Fill in the “good” intervals and read the answer off the number line.
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Answer: (−∞,−3] ∪ [6,∞)
![Page 23: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/23.jpg)
Method 2: Numerical (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
(x − 6)(x + 3) ≥ 0
x = −4 : (−4− 6)(−4 + 3) = (−10)(−1) = 10 ≥ 0
x = 0 : (0− 6)(0 + 3) = (−6)(3) = −18 ≥ 0
x = 7 : (7− 6)(7 + 3) = (1)(10) = 10 ≥ 0
6. Fill in the “good” intervals and read the answer off the number line.
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Answer: (−∞,−3] ∪ [6,∞)
![Page 24: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/24.jpg)
Examples
Solve the polynomial inequality and write the answer in interval notation:
1. 4x3 − 4x2 − x + 1 < 0
(−∞,−1
2
)∪(12 , 1
)2. x2 − x − 5 ≥ x − 2
(−∞,−1] ∪ [3,∞)
![Page 25: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/25.jpg)
Examples
Solve the polynomial inequality and write the answer in interval notation:
1. 4x3 − 4x2 − x + 1 < 0(−∞,−1
2
)∪(12 , 1
)
2. x2 − x − 5 ≥ x − 2
(−∞,−1] ∪ [3,∞)
![Page 26: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/26.jpg)
Examples
Solve the polynomial inequality and write the answer in interval notation:
1. 4x3 − 4x2 − x + 1 < 0(−∞,−1
2
)∪(12 , 1
)2. x2 − x − 5 ≥ x − 2
(−∞,−1] ∪ [3,∞)
![Page 27: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/27.jpg)
Examples
Solve the polynomial inequality and write the answer in interval notation:
1. 4x3 − 4x2 − x + 1 < 0(−∞,−1
2
)∪(12 , 1
)2. x2 − x − 5 ≥ x − 2
(−∞,−1] ∪ [3,∞)
![Page 28: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/28.jpg)
Rational Inequalities
![Page 29: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/29.jpg)
Method
Let’s work through the process using3x + 2
2 + x< 2.
1. Add/subtract terms to get zero on the right-hand-side (RHS) of theinequality and simplify to one fraction.
3x + 2
2 + x− 2 < 0
3x + 2
2 + x− 2·(2 + x)
1·(2 + x)< 0
3x + 2− 4− 2x
2 + x< 0
x − 2
x + 2< 0
![Page 30: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/30.jpg)
Method (continued)
2. Factor the top and bottom of the fraction.
x − 2
x + 2< 0
3. Set all factors on top and bottom equal to zero and solve.
Top: Bottom:
x − 2 = 0 x + 2 = 0x = 2 x = −2
![Page 31: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/31.jpg)
Method (continued)
2. Factor the top and bottom of the fraction.
x − 2
x + 2< 0
3. Set all factors on top and bottom equal to zero and solve.
Top: Bottom:
x − 2 = 0 x + 2 = 0x = 2 x = −2
![Page 32: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/32.jpg)
Method (continued)
4. Plot the values from step 3 on a number line and pick test pointson each side and between these numbers.
−4 −3 −2 −1 0 1 2 3 4
Test PointsFor values from the bottom, always use open circles.On top values, plot them with open circles for “<” or “>”.On top values, plot them with closed circles for “≤” or “≥”.
![Page 33: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/33.jpg)
Method (continued)
4. Plot the values from step 3 on a number line and pick test pointson each side and between these numbers.
−4 −3 −2 −1 0 1 2 3 4
Test Points
For values from the bottom, always use open circles.On top values, plot them with open circles for “<” or “>”.On top values, plot them with closed circles for “≤” or “≥”.
![Page 34: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/34.jpg)
Method (continued)
4. Plot the values from step 3 on a number line and pick test pointson each side and between these numbers.
−4 −3 −2 −1 0 1 2 3 4
Test Points
For values from the bottom, always use open circles.
On top values, plot them with open circles for “<” or “>”.On top values, plot them with closed circles for “≤” or “≥”.
![Page 35: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/35.jpg)
Method (continued)
4. Plot the values from step 3 on a number line and pick test pointson each side and between these numbers.
−4 −3 −2 −1 0 1 2 3 4
Test Points
For values from the bottom, always use open circles.On top values, plot them with open circles for “<” or “>”.
On top values, plot them with closed circles for “≤” or “≥”.
![Page 36: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/36.jpg)
Method (continued)
4. Plot the values from step 3 on a number line and pick test pointson each side and between these numbers.
−4 −3 −2 −1 0 1 2 3 4
Test Points
For values from the bottom, always use open circles.On top values, plot them with open circles for “<” or “>”.On top values, plot them with closed circles for “≤” or “≥”.
![Page 37: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/37.jpg)
Method (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
x − 2
x + 2< 0
x = −3 : −3−2−3+2 = −5
−1 = 5 < 0
x = 0 : 0−20+2 = −2
2 = −1 < 0
x = 3 : 3−23+2 = 1
5 < 0
6. Fill in the “good” intervals and read the answer off the number line.
−4 −3 −2 −1 0 1 2 3 4
Answer: (−2, 2)
![Page 38: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/38.jpg)
Method (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
x − 2
x + 2< 0
x = −3 : −3−2−3+2 = −5
−1 = 5 < 0
x = 0 : 0−20+2 = −2
2 = −1 < 0
x = 3 : 3−23+2 = 1
5 < 0
6. Fill in the “good” intervals and read the answer off the number line.
−4 −3 −2 −1 0 1 2 3 4
Answer: (−2, 2)
![Page 39: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/39.jpg)
Method (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
x − 2
x + 2< 0
x = −3 : −3−2−3+2 = −5
−1 = 5 < 0
x = 0 : 0−20+2 = −2
2 = −1 < 0
x = 3 : 3−23+2 = 1
5 < 0
6. Fill in the “good” intervals and read the answer off the number line.
−4 −3 −2 −1 0 1 2 3 4
Answer: (−2, 2)
![Page 40: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/40.jpg)
Method (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
x − 2
x + 2< 0
x = −3 : −3−2−3+2 = −5
−1 = 5 < 0
x = 0 : 0−20+2 = −2
2 = −1 < 0
x = 3 : 3−23+2 = 1
5 < 0
6. Fill in the “good” intervals and read the answer off the number line.
−4 −3 −2 −1 0 1 2 3 4
Answer: (−2, 2)
![Page 41: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/41.jpg)
Method (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
x − 2
x + 2< 0
x = −3 : −3−2−3+2 = −5
−1 = 5 < 0
x = 0 : 0−20+2 = −2
2 = −1 < 0
x = 3 : 3−23+2 = 1
5 < 0
6. Fill in the “good” intervals and read the answer off the number line.
−4 −3 −2 −1 0 1 2 3 4
Answer: (−2, 2)
![Page 42: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/42.jpg)
Method (continued)
5. Check to see if each interval is “good” or “bad” using the testpoints.
x − 2
x + 2< 0
x = −3 : −3−2−3+2 = −5
−1 = 5 < 0
x = 0 : 0−20+2 = −2
2 = −1 < 0
x = 3 : 3−23+2 = 1
5 < 0
6. Fill in the “good” intervals and read the answer off the number line.
−4 −3 −2 −1 0 1 2 3 4
Answer: (−2, 2)
![Page 43: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/43.jpg)
Example
Solve the polynomial inequality and write the answer in interval notation:
1.x − 3
x + 4≤ x + 2
x − 5
(−4, 12
]∪ (5,∞)
2.(x − 3)2(x + 1)
(x − 5)≥ 0
(−∞,−1] ∪ {3} ∪ (5,∞)
![Page 44: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/44.jpg)
Example
Solve the polynomial inequality and write the answer in interval notation:
1.x − 3
x + 4≤ x + 2
x − 5(−4, 12
]∪ (5,∞)
2.(x − 3)2(x + 1)
(x − 5)≥ 0
(−∞,−1] ∪ {3} ∪ (5,∞)
![Page 45: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/45.jpg)
Example
Solve the polynomial inequality and write the answer in interval notation:
1.x − 3
x + 4≤ x + 2
x − 5(−4, 12
]∪ (5,∞)
2.(x − 3)2(x + 1)
(x − 5)≥ 0
(−∞,−1] ∪ {3} ∪ (5,∞)
![Page 46: Section 4.6 - Polynomial Inequalities and Rational Inequalitiesain.faculty.unlv.edu/124 Notes/Chapter 4/Section 4.6... · 2018. 8. 23. · Let’s work through the process using (x](https://reader035.vdocuments.mx/reader035/viewer/2022081407/6056198ad756cb1995705667/html5/thumbnails/46.jpg)
Example
Solve the polynomial inequality and write the answer in interval notation:
1.x − 3
x + 4≤ x + 2
x − 5(−4, 12
]∪ (5,∞)
2.(x − 3)2(x + 1)
(x − 5)≥ 0
(−∞,−1] ∪ {3} ∪ (5,∞)