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MT7 Test Notes Working with Rationals

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MT7: Working with Rationals 20 questions that cover Adding, Subtracting (like denominators and unlike denominators), Multiplying, and Dividing. This is a longer test that will be difficult! Study these two pages well. Notes are provided at the bottom of each page.

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Notes: Adding a subtracting Rational with like denominators is the easiest section of this test. The mistake that many students make is adding and subtracting the integers. MT7: Working with RationalsTest Notes Same denominator? Too Easy! CLT: (5x + x) + (4y – y) 24y 2 6x + 3y 24y 2 Now decide: Can you GCF? Diamond? Diff of 2 Squares? GCF! 3(2x + y) 24y 2 Now reduce: (2x + y) 8y 2 Remember: You can never, ever, ever cancel into parenthesis. That is why the 2 and 8 cannot be reduced. CLT: (x + x) + (y + 6y) 25xy 2x + 7y 25xy Now decide: Can you GCF? Diamond? Diff of 2 Squares? No! All done. (Nothing goes into 2 and 7)

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MT7: Working with Rationals 20 questions that cover Adding, Subtracting (like denominators and unlike denominators), Multiplying, and Dividing. This is a longer test that will be difficult! Study these two pages well. Notes are provided at the bottom of each page.

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Notes: Adding a subtracting Rational with like denominators is the easiest section of this test. The mistake that many students make is just adding and subtracting the integers. MT7: Working with RationalsTest Notes More of the same CLT: (2v – 4v) + (2u) 10uv 3 -2v + 2u 10uv 3 Now decide: Can you GCF? Diamond? Diff of 2 Squares? GCF! -2(v - u) 10uv 3 Now reduce: v – u 5uv 3 Notice the sign change of the u? That happened because we pulled out a “-”. CLT: (x + x) + (-6y + y) 4xy 2x – 5y 4xy Now decide: Can you GCF? Diamond? Diff of 2 Squares? No! All done. (Nothing goes into 2 and 5)

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MT7: Working with Rationals This section is the hardest part of the test. Make sure you can find a common denominator.

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Notes: Must find common denominator first. A key is to take one of everything you see and take the largest of something if there is more than one. Making a box to put your extra numbers in can be helpful, but is not required. MT7: Working with RationalsTest Notes Not common Denominators! (r+6)(r-2) (r-2) (r+6) How do I know what goes here? 4r(r-2) + (r+6)(r+6) (r+6)(r-2) 4r 2 -8r + r 2 +12r+36 (r+6)(r-2) 5r 2 +4r+36 (3n)(10n+2) (10n+2) 3n 2n(10n+2) + 5(3n) (3n)(10n+2) 20n 2 + 4n + 15n (3n)(10n+2) 20n 2 +19n Now GCF and reduce (3n)(10n+2) n(20n+19) 3(10n+2) (20n+19)

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MT7: Working with Rationals This section is the hardest part of the test. Make sure you can find a common denominator.

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Notes: Must find common denominator first. A key is to take one of everything you see and take the largest of something if there is more than one. Making a box to put your extra numbers in can be helpful, but is not required. MT7: Working with RationalsTest Notes (3x)(x+5) (x+5) (3x) (x-3)(x+5) - 3x(x-1) (3x)(x+5) x 2 +2x-15-3x 2 +3x (3x)(x+5) -2x 2 + 5x – 15 (m+6)(m+5) (m+5) (m+6) 3m(m+5) - 6(m+6) (m+6)(m+5) 3m 2 +15m-6m-36 (m+6)(m+5) 3m 2 +9m-36 No GCF, Diamond, or Diff of 2 Squares! All Done. Foil and Distribute Watch out distributing “-” 3, 9, 36? Time to GCF (m+6)(m+5) 3(m 2 +3m-12)

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MT7: Working with Rationals This is from 7.3. Pretty easy as long as you can factor.

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Notes: No need to find common denominator (not adding or subtracting). Just find the right method to factor and cancel your final results when possible. MT7: Working with RationalsTest Notes 24 -11 Look at the pattern! Diamond Method (b )(b ) -8 -3 -27 6 9 -3 (b )(b ) + 9 - 3 Now just cancel. (b - 8) (b + 9) -12 (v )(v ) -4 3 - 4 + 3 -24 -5 -8 3 (v )(v ) - 8 + 3 Now just cancel. (v - 4) (v - 8)

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MT7: Working with Rationals This is from 7.3. Pretty easy as long as you can factor.

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Notes: No need to find common denominator (not adding or subtracting). Just find the right method to factor and cancel your final results when possible. MT7: Working with RationalsTest Notes Look at the pattern! Difference of 2 Squares and Diamond (n )(n ) - 9 + 9 -36 -5 -9 4 (n )(n ) - 9 + 4 Now just cancel. (n + 9) (n + 4) 36 -13 (x )(x ) -9 -4 12 -7 -4 -3 (x )(x ) - 4 - 3 Now just cancel. (x - 9) (x - 3)

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MT7: Working with Rationals Similar to 7.3. Pretty easy as long as you can factor and reduce.

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Notes: Almost like 7.3. Look for ways to factor then cancel. Watch out for ÷! MT7: Working with RationalsTest Notes Look for GCF, Diamond, or Diff of 2 Squares. 1 8m 2 Now just cancel. 1414 ● 2(m – 8) (m – 8) GCF! 1 4 Divide means “Flip” 5r-35 4 ● r + 4 5r - 35 Now just cancel. r + 4 4

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MT7: Working with Rationals Similar to 7.3. Pretty easy as long as you can factor and reduce.

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Notes: Almost like 7.3. Look for ways to factor then cancel. Watch out for ÷! MT7: Working with RationalsTest Notes Look for GCF, Diamond, or Diff of 2 Squares. b+2 b 2 – 5b - 14 Now just cancel. 2b 2 (b-7) ● 2b 2 1 Divide means “Flip” b+2 (b – 7)(b + 2) ● 2b 2 1 Now just cancel. 5 5(n – 2) ● (n + 4)(n – 2) 2 (n + 4) 2

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MT7: Working with Rationals The section on solving rationals is actually easier than adding them because you get to cancel the denominator after you find LCD.

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Notes: Almost like 7.52 only after you find your LCD, you get to erase it. Don’t forget to solve your equations after you delete your denominator! MT7: Working with RationalsTest Notes 3x 2 Common denominator between x, 3x 2, and 3x 2 ? Must find LCD first… Now follow the process… 3x 1 1 3x = 1 + x + 1 Delete denominator 3x = 1 + x + 1 Solve for x 2x = 2 x = 1 6k Common denominator between 3, 6, and 6k? 4k = k + k - 4 4k = 2k - 4 2k = -4 2k k 1 k = -2

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MT7: Working with Rationals The section on solving rationals is actually easier than adding them because you get to cancel the denominator after you find LCD.

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Notes: Almost like 7.52 only after you find your LCD, you get to erase it. Don’t forget to solve your equations after you delete your denominator! MT7: Working with RationalsTest Notes 6n 2 Common denominator between 6n, n 2, and 6n 2 ? Same thing… Now follow the process… n 6 1 n = 6 - n - 2 Watch for “-”! n = 6 - n - 2 Solve for x 2n= 4 x = 2 4v 2 Common denominator between 2v 2, 4v 2, and v 2 ? 2 + v – 4 = 4 v – 2 = 4 v = 6 2 1 4

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That’s your entire test. Make sure you don’t make the easy mistakes. The easy mistakes are usually messing up when you add, subtract, multiply and divide. Your precision is going to matter.

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