Topic Review — Linear Equations and Inequalities — Solutions

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1. Solving linear equations

   1. 3x + 7 = 22: 3x = 15 → x = 5
   2. 5(y − 2) = 15: y − 2 = 3 → y = 5
   3. 4m − 3 = 2m + 9: 2m = 12 → m = 6
   4. −3(2k + 1) = 9: −6k − 3 = 9 → −6k = 12 → k = −2
   5. 7 − 2x = −5: −2x = −12 → x = 6

2. Equations with fractions

   1. (x + 3)/4 = 5: x + 3 = 20 → x = 17
   2. x/2 + x/5 = 7: LCD=10: 5x + 2x = 70 → 7x = 70 → x = 10
   3. (3y − 1)/2 = (y + 5)/3: Cross-multiply: 3(3y−1) = 2(y+5) → 9y−3 = 2y+10 → 7y = 13 → y = 13/7
   4. 4/x = 8: 4 = 8x → x = 1/2
   5. x/3 − x/4 = 2: LCD=12: 4x − 3x = 24 → x = 24

3. Linear inequalities

   1. 2x − 5 > 3: 2x > 8 → x > 4. Open circle at 4, arrow right.
   2. −3y + 6 ≤ 0: −3y ≤ −6 → flip: y ≥ 2. Closed circle at 2, arrow right.
   3. 4m + 1 ≥ 2m − 7: 2m ≥ −8 → m ≥ −4
   4. 5 − (x + 1) < 2x − 2: 4 − x < 2x − 2 → 6 < 3x → x > 2
   5. −3 ≤ 2x + 1 < 9: Subtract 1: −4 ≤ 2x < 8. Divide by 2: −2 ≤ x < 4. Closed circle at −2, open circle at 4, segment between.

4. Mixed equations and forming algebraic expressions

   1. Consecutive even integers sum = 54: n + (n+2) + (n+4) = 54 → 3n + 6 = 54 → 3n = 48 → n = 16. Integers: 16, 18, 20
   2. (2x + 1)/3 + (x − 2)/4 = 3: LCD=12. 4(2x+1) + 3(x−2) = 36 → 8x+4+3x−6 = 36 → 11x−2 = 36 → 11x = 38 → x = 38/11
   3. Student error in −2x + 4 > 10: −2x > 6 → divide by −2, flip: x < −3. The student wrote x > −3 without flipping the sign when dividing by a negative. Correct answer: x < −3.
   4. Is x = 4 a solution to 3x − 2 ≥ 5x − 10?: Sub x=4: LHS = 10, RHS = 10. 10 ≥ 10. Yes, x = 4 satisfies the inequality (just barely, since it gives equality).

5. Problem solving

   1. Yuki and Ben saving: 40 + 15w = 100 + 8w → 7w = 60 → w = 60/7 ≈ 8.57. They have the same amount after 8&frac{4}{7} weeks, i.e. during the 9th week. (At w=9: Yuki=$175, Ben=$172; at w=8: Yuki=$160, Ben=$164. They are equal at exactly w = 60/7 weeks.)
   2. Rectangular paddock, P=86 m, l = 2w + 7: 2(l + w) = 86 → l + w = 43. Sub l = 2w+7: 2w+7+w = 43 → 3w = 36 → w = 12. l = 2(12)+7 = 31. Width = 12 m, Length = 31 m
   3. Phone plan $12 + $0.08t ≤ $30: 0.08t ≤ 18 → t ≤ 225. Maximum = 225 texts
   4. Mei needs average ≥ 75% over 4 assessments, scored 70, 80, 68: (70 + 80 + 68 + s)/4 ≥ 75 → 218 + s ≥ 300 → s ≥ 82. Mei needs at least 82% on her final assessment.