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Absonsulte Values and Absolute Inequalities Notes
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Absolute value notes Day 1 1. Absolute value of a number is the distance from 0 on a number line. It represents positive distance without direction. → The absolute value function is defined by a piecewise function. O 1x1 = O x if x ≥ 0 2. Properties of Aboslute Value Parent Function. f(x) = 1x1 6 -xif x20 Negative Slope ↳ Parent function is the "basic version" Domain: (-∞∞) • Vertex: (0,0) Range: [0, ∞0) two-part. always all real numbers. Starts at vertex then goes up or down. always at a point Axis of Symmetry • X=0 vertical line through center of graph Interval of Increase: (0,∞) interval of x-values where slope is positive Interval of decrease: (-∞,0) interval of x-values where slope is negative Positive Slope X 3. Absolute value Transformation Function • g(x) = A/B (x-C)| +D * The vertex is (C₁D) * Note: there is no f(x) in the equation, only absolute value! called an explicit equation. 4. Graphing Absolute Valve with Transformations. Graphing Method #1: Transformations. (-4,1) (41) |(0,0) 1. Use the parent function points. 4 (0,0), (1₁1) and (-1, 1) 2. Use transformation rules to graph the example 4 g(x) = -4/2x+4/+1 ↳ FACTOR! + g(x) = -4/2(x+2) | + +1₁ Method #1: Write a role to find the image Points (1/2 x-2₁-4y+¹) (1½/₂2 (0)-2, -4(0)+1) (-2,1) (1/2₂ (1)-2₁ - 4(1)+1) → (-1.5, -3) (1/2 (-1)-2₁-4 (1)+1) (-2.5,-3) "New Vertex" using defination of 1x1 to solve equation. Recall that absolute value is distance from 0. 3. 1x1=3 X=3 -X=3 x=3,-3 Ik-31-10 K-3-10 Absolute values Equations (Day 2) -(1-3)=10 K-3=-10 K=13₁-7 (-4,4) u Check:...
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Alternative transcript:
12-6)+4/-66-6) +257 1-817-8x -2 10 2x+4 -3 which values on the number line. are 3 units from 0?! 52+x9 + -1 0 I 2 3 Analyzing graphs of Ixl to find Solutions. Solutions are where the absolute value graph intercests the line.. 12x+41=6x + 28 2x+4= 6x+28 -4x=24 -4 -4 X=-6 Extranedus solution 10 13 Check Answers. 113-31=1101=10✓ 1-7-31=1-101 = 10r - (2x+4)= 6x+28 -2x -4 = 28 +6x -32= 8x -42X Check: 12(-4) +41 26 (-4) +28 1-41=4✓ TO Solve an absolute valve equation.. 1. Isolate the absolute value 2. Set up two cases: 4 Positive: Drop the absolute value bars and sowe for x OR! Negative: Put a negative in front, replace the absolute. value bars with Paraentheses, and solue for x 3. Plug solutions back into the equation and check. for extraneous solutions. This is required if there are variables are outside of the absolute value. 1 Solve the equation (Isolate the absolute valve bars first!!) 141-6--2 +6 1₁1=4 POSITIVE 4 . = 4.4 x = 16 1X = ± 16 4. X INEGITIVE - (2)=4 -4.4 = X = -16 2. 3. POSITIVE X+9=13 -9-9 x=4 POSITUTUE 4-3x=2x+3 +3x+3x -3 {+x5=h Je don't add! 5x=1 -3 2 = 4. √x +5 lis the inside and if an absolute on value is on the inside like that x+5=1 -5 -5 X=-4 1x +91=13 1x=4 or x=-22 14-3x1 = 2x +3 X=0r x=7 NEGATIVE = -(x+9)=13 -X-9=13 +9+9 1x=-4₁-6 -X=22 -XX same like terms +4 X=-22 NEGATIVE (4-3x) -4+3x = 2x + 3 2x+3 2x - 4 + x = = X=1 31x+5/+12 71x +51 +8 -7 -4/x+51 +1² = 8 -12 -4/x+51=-4x+5/-1 3 +4 -X=6 - -(x+5)=1 -X-5=1 +5+5 9=× = x ² ==== G Challenge Question 1! NEGATIVE 17x-31=13x+71 7x-3-(3x+7) = 7x-3--3x-7 +3x +3x 10x-3=-7 +3 +3 10X=-4-2 105 10 x=-25 Challenge Question 2! 1 POSITIVEL X+2>0 X+2≤4 x≤4-2 x≤2 1-6≤x≤2 -2 x= -4/5 or x = 5/1/2 1x +21=4 4 POSITIVE] 7x-3= 3x +7 - 3x - 3x 4x=3 =1 +3 +3 4x = 10 4 4 x=¾/2 NEGATIVE 072+x - (x+2) = 4 X+2=-4 X≥-4 -2 9-=X - LIN 5 1. Write an absolute value equation that would have no Solutions 1x1=-6 Absolute value can never be equal to a negative! 2. what about one that has all real numbers as the Solution? 1x1 = |x| Ex:1. 1x13 which values are within 3 units from 03 1-34x23 3-2-1 0 Ex:21x123 -4 x≤-3 which values are 3 or more units away from 0? Ex:3 14x-8/>12 Absolute Value Inequalities. 4х-8>12 4x>20 x>5 -5-4-3-2 3 - 2 4 or 0 1 2 x231 34 3 4 -(4x-8)>12 4x -8<-12 → Flip sign 4x<-4 X<-1 Possible Answers: X2-100 x 75 •WAYYYYY... (-∞, -1) U (5,∞0) | more simple Types of Absolute Value Inequalities! Inequality туре AND OR Absolute Value Inequality lax+b/²c 1ax +bl≤c lax+b|>c /ax+b/2c Compound Inequality -c<ax+b²c -(≤ax+b≤c ax+bc-cor ax+b>c axtb ≤ c or ax+b=l Number line REMEMBER TO ISOLATE THE ABSOLUTE VALUE FIRST! → To Solve an absolute value inequality, 1. Isolate the absolute value 2. Set up two cases: A. Positive: Drop the absolute value bars and solve the inequality B. Negative: Put a negative in front, replace absolute value bars. with parentheses, and solve the inequality. Reminders: A. Flip the sign when dividing or multiplying both sides by a negative. B. An absolute value cannot be less than a negative number.. C. An absolute value is always greater than a negative number. Practice !!! Solution 1: -49+2=10 -2 -4928 9=-2 Solution 2 2x+7≤37 2x≤-4 -2 12x+71 ≤ 1².3 3 2 2 1x≤ -2] • 1-4q+21= 10 Possible Answers: 9/2-2 or 9/23 (-∞0₁ -2] u [3,00) -2 laiser! 12x+71 3 2 1 Possible ANSWERS x²-2 and x = -5 [-5-2] 1-(-49+2) = 10 -4q+2 ≤-10 -2 -2 -49≤-12 -4 923 - (2x + 7) =≤ 3 -2x-7≤3 +7 +7 -2x≤10 -2 -2 |X2-5 I ) Solution 3 0.5ral O.T Trez Solution 4. 0.5871 r>2 Solution 5 10.5r1-42-3 +4 10.50/²1 Possible ANSWER -22122 +4 10·51-4-3 +4 +4 70-58131 POSSIBLE ANSWER. r²-2 or r> 2 2 -(0-5)41 -0.5721 -0.5-0.5 1r7-2 ا (0) - r²-2 -1/2/p-2/23 NO SOLUTIONS! Absolute value cannot be less than a negative number. L T
Absonsulte Values and Absolute Inequalities Notes
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Asna Kadiwal
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Solving and writing absolute values
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Absonsulte Values and Absolute Inequalities Notes
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Solving and writing absolute values
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Absolute value notes Day 1 1. Absolute value of a number is the distance from 0 on a number line. It represents positive distance without direction. → The absolute value function is defined by a piecewise function. O 1x1 = O x if x ≥ 0 2. Properties of Aboslute Value Parent Function. f(x) = 1x1 6 -xif x20 Negative Slope ↳ Parent function is the "basic version" Domain: (-∞∞) • Vertex: (0,0) Range: [0, ∞0) two-part. always all real numbers. Starts at vertex then goes up or down. always at a point Axis of Symmetry • X=0 vertical line through center of graph Interval of Increase: (0,∞) interval of x-values where slope is positive Interval of decrease: (-∞,0) interval of x-values where slope is negative Positive Slope X 3. Absolute value Transformation Function • g(x) = A/B (x-C)| +D * The vertex is (C₁D) * Note: there is no f(x) in the equation, only absolute value! called an explicit equation. 4. Graphing Absolute Valve with Transformations. Graphing Method #1: Transformations. (-4,1) (41) |(0,0) 1. Use the parent function points. 4 (0,0), (1₁1) and (-1, 1) 2. Use transformation rules to graph the example 4 g(x) = -4/2x+4/+1 ↳ FACTOR! + g(x) = -4/2(x+2) | + +1₁ Method #1: Write a role to find the image Points (1/2 x-2₁-4y+¹) (1½/₂2 (0)-2, -4(0)+1) (-2,1) (1/2₂ (1)-2₁ - 4(1)+1) → (-1.5, -3) (1/2 (-1)-2₁-4 (1)+1) (-2.5,-3) "New Vertex" using defination of 1x1 to solve equation. Recall that absolute value is distance from 0. 3. 1x1=3 X=3 -X=3 x=3,-3 Ik-31-10 K-3-10 Absolute values Equations (Day 2) -(1-3)=10 K-3=-10 K=13₁-7 (-4,4) u Check:...
Absolute value notes Day 1 1. Absolute value of a number is the distance from 0 on a number line. It represents positive distance without direction. → The absolute value function is defined by a piecewise function. O 1x1 = O x if x ≥ 0 2. Properties of Aboslute Value Parent Function. f(x) = 1x1 6 -xif x20 Negative Slope ↳ Parent function is the "basic version" Domain: (-∞∞) • Vertex: (0,0) Range: [0, ∞0) two-part. always all real numbers. Starts at vertex then goes up or down. always at a point Axis of Symmetry • X=0 vertical line through center of graph Interval of Increase: (0,∞) interval of x-values where slope is positive Interval of decrease: (-∞,0) interval of x-values where slope is negative Positive Slope X 3. Absolute value Transformation Function • g(x) = A/B (x-C)| +D * The vertex is (C₁D) * Note: there is no f(x) in the equation, only absolute value! called an explicit equation. 4. Graphing Absolute Valve with Transformations. Graphing Method #1: Transformations. (-4,1) (41) |(0,0) 1. Use the parent function points. 4 (0,0), (1₁1) and (-1, 1) 2. Use transformation rules to graph the example 4 g(x) = -4/2x+4/+1 ↳ FACTOR! + g(x) = -4/2(x+2) | + +1₁ Method #1: Write a role to find the image Points (1/2 x-2₁-4y+¹) (1½/₂2 (0)-2, -4(0)+1) (-2,1) (1/2₂ (1)-2₁ - 4(1)+1) → (-1.5, -3) (1/2 (-1)-2₁-4 (1)+1) (-2.5,-3) "New Vertex" using defination of 1x1 to solve equation. Recall that absolute value is distance from 0. 3. 1x1=3 X=3 -X=3 x=3,-3 Ik-31-10 K-3-10 Absolute values Equations (Day 2) -(1-3)=10 K-3=-10 K=13₁-7 (-4,4) u Check:...
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Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.
Still not sure? Look at what your fellow peers are saying...
iOS User
I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D
Stefan S, iOS User
The application is very simple and well designed. So far I have found what I was looking for :D
SuSSan, iOS User
Love this App ❤️, I use it basically all the time whenever I'm studying
Alternative transcript:
12-6)+4/-66-6) +257 1-817-8x -2 10 2x+4 -3 which values on the number line. are 3 units from 0?! 52+x9 + -1 0 I 2 3 Analyzing graphs of Ixl to find Solutions. Solutions are where the absolute value graph intercests the line.. 12x+41=6x + 28 2x+4= 6x+28 -4x=24 -4 -4 X=-6 Extranedus solution 10 13 Check Answers. 113-31=1101=10✓ 1-7-31=1-101 = 10r - (2x+4)= 6x+28 -2x -4 = 28 +6x -32= 8x -42X Check: 12(-4) +41 26 (-4) +28 1-41=4✓ TO Solve an absolute valve equation.. 1. Isolate the absolute value 2. Set up two cases: 4 Positive: Drop the absolute value bars and sowe for x OR! Negative: Put a negative in front, replace the absolute. value bars with Paraentheses, and solue for x 3. Plug solutions back into the equation and check. for extraneous solutions. This is required if there are variables are outside of the absolute value. 1 Solve the equation (Isolate the absolute valve bars first!!) 141-6--2 +6 1₁1=4 POSITIVE 4 . = 4.4 x = 16 1X = ± 16 4. X INEGITIVE - (2)=4 -4.4 = X = -16 2. 3. POSITIVE X+9=13 -9-9 x=4 POSITUTUE 4-3x=2x+3 +3x+3x -3 {+x5=h Je don't add! 5x=1 -3 2 = 4. √x +5 lis the inside and if an absolute on value is on the inside like that x+5=1 -5 -5 X=-4 1x +91=13 1x=4 or x=-22 14-3x1 = 2x +3 X=0r x=7 NEGATIVE = -(x+9)=13 -X-9=13 +9+9 1x=-4₁-6 -X=22 -XX same like terms +4 X=-22 NEGATIVE (4-3x) -4+3x = 2x + 3 2x+3 2x - 4 + x = = X=1 31x+5/+12 71x +51 +8 -7 -4/x+51 +1² = 8 -12 -4/x+51=-4x+5/-1 3 +4 -X=6 - -(x+5)=1 -X-5=1 +5+5 9=× = x ² ==== G Challenge Question 1! NEGATIVE 17x-31=13x+71 7x-3-(3x+7) = 7x-3--3x-7 +3x +3x 10x-3=-7 +3 +3 10X=-4-2 105 10 x=-25 Challenge Question 2! 1 POSITIVEL X+2>0 X+2≤4 x≤4-2 x≤2 1-6≤x≤2 -2 x= -4/5 or x = 5/1/2 1x +21=4 4 POSITIVE] 7x-3= 3x +7 - 3x - 3x 4x=3 =1 +3 +3 4x = 10 4 4 x=¾/2 NEGATIVE 072+x - (x+2) = 4 X+2=-4 X≥-4 -2 9-=X - LIN 5 1. Write an absolute value equation that would have no Solutions 1x1=-6 Absolute value can never be equal to a negative! 2. what about one that has all real numbers as the Solution? 1x1 = |x| Ex:1. 1x13 which values are within 3 units from 03 1-34x23 3-2-1 0 Ex:21x123 -4 x≤-3 which values are 3 or more units away from 0? Ex:3 14x-8/>12 Absolute Value Inequalities. 4х-8>12 4x>20 x>5 -5-4-3-2 3 - 2 4 or 0 1 2 x231 34 3 4 -(4x-8)>12 4x -8<-12 → Flip sign 4x<-4 X<-1 Possible Answers: X2-100 x 75 •WAYYYYY... (-∞, -1) U (5,∞0) | more simple Types of Absolute Value Inequalities! Inequality туре AND OR Absolute Value Inequality lax+b/²c 1ax +bl≤c lax+b|>c /ax+b/2c Compound Inequality -c<ax+b²c -(≤ax+b≤c ax+bc-cor ax+b>c axtb ≤ c or ax+b=l Number line REMEMBER TO ISOLATE THE ABSOLUTE VALUE FIRST! → To Solve an absolute value inequality, 1. Isolate the absolute value 2. Set up two cases: A. Positive: Drop the absolute value bars and solve the inequality B. Negative: Put a negative in front, replace absolute value bars. with parentheses, and solve the inequality. Reminders: A. Flip the sign when dividing or multiplying both sides by a negative. B. An absolute value cannot be less than a negative number.. C. An absolute value is always greater than a negative number. Practice !!! Solution 1: -49+2=10 -2 -4928 9=-2 Solution 2 2x+7≤37 2x≤-4 -2 12x+71 ≤ 1².3 3 2 2 1x≤ -2] • 1-4q+21= 10 Possible Answers: 9/2-2 or 9/23 (-∞0₁ -2] u [3,00) -2 laiser! 12x+71 3 2 1 Possible ANSWERS x²-2 and x = -5 [-5-2] 1-(-49+2) = 10 -4q+2 ≤-10 -2 -2 -49≤-12 -4 923 - (2x + 7) =≤ 3 -2x-7≤3 +7 +7 -2x≤10 -2 -2 |X2-5 I ) Solution 3 0.5ral O.T Trez Solution 4. 0.5871 r>2 Solution 5 10.5r1-42-3 +4 10.50/²1 Possible ANSWER -22122 +4 10·51-4-3 +4 +4 70-58131 POSSIBLE ANSWER. r²-2 or r> 2 2 -(0-5)41 -0.5721 -0.5-0.5 1r7-2 ا (0) - r²-2 -1/2/p-2/23 NO SOLUTIONS! Absolute value cannot be less than a negative number. L T